Math Final Exam
(0, -1)
3π/2
Appropriate
Draw a diagram, where...
Highest & Lowest
The _____ points on a graph indicate the maxima and minima.
Sum Formula For Cosine
cos(α+β)=cos α cos β−sin αsin β
Reciprocal Function
f(x)=1 / x
f(x)
k is a zero of f(x) if and only if (x−k) is a factor of...
Sum Formula for Sine
sin(α+β)=sin α cos β+cos α sin β
The Quadratic Formula
x=(-b+ or - sqrt(b^2-4ac)) / 2a.
Shifted, Compressed, And/Or Stretched Cotangent Function
y=Acot(Bx-C)+D
(-1, 0)
π
(0, 1)
π/2
(1/2, sqrt(3)/2)
π/3
The Pythagorean Theorem
...along with the sum and difference formulas can be used to find multiple sums and differences of angles.
Horizontal Lines
...are written in the form f(x)=b.
Vertical Lines
...are written in the form x=b.
Natural Logarithms
...can be evaluated using a calculator.
The Vertex & the Intercepts
...can be identified and interpreted to solve real-world problems.
Interpolation, Extrapolation
...can be used to predict values inside the domain and range of the data, whereas _____ can be used to predict values outside the domain and range of the data.
Application
...problems are often easier to solve by using sum and difference formulas.
Odd Functions
...satisfy the condition f(x)=-f(-x).
Even Functions
...satisfy the condition f(x)=f(-x).
Scatter Plots
...show the relationship between two sets of data.
Approximate
...solutions of the equation f(x)=b^(x+c)+d can be found using a graphing calculator.
Applied Problems
...such as ranges of possible values, can also be solved using the absolute value function.
Identifying Points
...that mark the interval on a graph can be used to find the average rate of change.
(1, 0)
0, 2π
(1/2, -sqrt(3)/2)
5π/3
(-sqrt(2)/2, -sqrt(2)/2)
5π/4
(-sqrt(3)/2, 1/2)
5π/6
(-sqrt(3)/2, -1/2)
7π/6
Trigonometric Function
A calculator will return an angle within the restricted domain of the original...
Complex Plane
A coordinate system in which the horizontal axis is used to represent the real part of a complex number and the vertical axis is used to represent the imaginary part of a complex number.
Pythagorean Identity
A corollary of the Pythagorean Theorem stating that the square of the cosine of a given angle plus the square of the sine of that angle equals 1. ((cos^2)t + (sin^2)t = 1)
Horizontal Shift
A function can also be graphed by identifying its amplitude, period, phase shift, and... / A transformation that shifts a function's graph left or right by adding a positive or negative constant to the input. (g(x)=f(x-h) (right for h>0)).
Horizontally
A function can be compressed or stretched _____ by multiplying the input by a constant.
Vertically
A function can be compressed or stretched _____ by multiplying the output by a constant.
Neither
A function can be odd, even, or...
Periodic Function
A function f(x) that satisfies f(x+P)=f(x) for a specific constant P and any value of x.
One to One Function
A function for which each value of the output is associated with a unique input value.
Piecewise Function
A function in which more than one formula is used to define the output.
Increasing, Decreasing
A function is _____ where its rate of change is positive and _____ where its rate of change is negative.
Decreasing Function
A function is decreasing in some open interval if f(b)<f(a) for any two input values a and b in the given interval where b>a.
Specific Value
A function is evaluated by solving at a...
Increasing Function
A function is increasing in some open interval if f(b)>f(a) for any two input values a and b in the given interval where b>a.
Only One
A function is one to one if each output value corresponds to _____ input value.
f(-x)=-f(x)
A function is said to be even if f(−x)=f(x) and odd if...
Logistic Growth Model
A function of the form f(x)=c/(1+ae^(−bx)) where c/(1+a) is the initial value, c is the carrying capacity, or limiting value, and b is a constant determined by the rate of growth.
One At a Time
A function presented as an equation can be reflected by applying transformations...
Power Function
A function that can be represented in the form f(x)=kx^p where k is a constant, the base is a variable, and the exponent, p, is a constant.
Rational Function
A function that can be written as the ratio of two polynomials. (f(x)=P(x)/Q(x)=(apx^p +a(p-1)x^(p-1) +...+a1x+a0)/(bqx^q +b(q-1)x^(q-1) +...+b1x+b0), Q(x) does not equal 0)
Polynomial Function
A function that consists of either zero or the sum of a finite number of non-zero terms, each of which is a product of a number, called the coefficient of the term, and a variable raised to a non-negative integer power. / Graphing a _____ helps to estimate local and global extremas. / Synthetic division can be used to find the zeros of a...
More Than One
A function that levels off at a horizontal value has a horizontal asymptote. A function can have _____ vertical asymptote. / A piecewise function is described by _____ formula.
Vertical Compression
A function transformation that compresses the function's graph vertically by multiplying the output by a constant 0<a<1. (g(x)=af(x)(0<a<1)).
Continuous Function
A function whose graph can be drawn without lifting the pen from the paper because there are no breaks in the graph.
Odd Function
A function whose graph is unchanged by combined horizontal and vertical reflection, f(x)=-f(-x), and is symmetric about the origin.
Even Function
A function whose graph is unchanged by horizontal reflection, f(x)=f(-x), and is symmetric about the y axis.
Linear Function
A function with a constant rate of change that is a polynomial of degree 1, and whose graph is a straight line.
Decreasing Linear Function
A function with a negative slope. If f(x)=mx+b, then m<0.
Increasing Linear Function
A function with a positive slope. If f(x)=mx+b, then m>0.
Does Not
A graph can be reflected both vertically and horizontally. The order in which the reflections are applied _____ affect the final graph.
Below the Other Line
A graph of the system may be used to identify the points where one falls...
Vertical Line
A graph represents a function if any _____ drawn on the graph intersects the graph at no more than one point. / A line defined by x=a, where a is a real number. The slope of a vertical line is undefined.
Horizontal Asymptote
A horizontal line y=b where the graph approaches the line as the inputs increase or decrease without bound.
Negative Reciprocal Slope
A line perpendicular to another line, passing through a given point, may be found in the same manner, with the exception of using the...
Larger
A local maximum is where a function changes from increasing to decreasing and has an output value ______ than output values at neighboring input values.
Smaller
A local minimum is where the function changes from decreasing to increasing and has an output value ______ than output values at neighboring input values.
Interval Notation
A method of describing a set that includes all numbers between a lower limit and an upper limit. The lower and upper values are listed between brackets or parentheses, a square bracket indicating inclusion in the set, and a parenthesis indicating exclusion.
Horizontal Line Test
A method of testing whether a function is one to one by determining whether any horizontal line intersects the graph more than once.
Vertical Line Test
A method of testing whether a graph represents a function by determining whether a vertical line intersects the graph no more than once.
Coefficient
A nonzero real number multiplied by a variable raised to an exponent.
Imaginary Number
A number in the form bi where i=sqrt(−1).
Assigned Subdomain
A piecewise function can be graphed using each algebraic formula on its...
Positive Whole Number Power
A polynomial function is the sum of terms, each of which consists of a transformed power function with...
Two
A polynomial function of degree _____ is called a quadratic function. / Polynomial functions of degree _____ or more are smooth, continuous functions. / An exponential model can be found when the _____ data points from the model are known.
Fundamental Theorem of Algebra
A polynomial function with degree greater than 0 has at least one complex 0.
n-1
A polynomial of degree n will have at most n x-intercepts and at most _____ turning points. / A polynomial function of degree n has at most _____ turning points.
Number Power
A power function is a variable base raised to a...
Y Value
A quadratic function's minimum or maximum value is given by the _____ of the vertex.
Input Quantity
A rate of change relates a change in an output quantity to a change in an _____. The average rate of change is determined using only the beginning and ending data.
Leading Terms
A rational function's end behavior will mirror that of the ratio of the _____ of the numerator and denominator functions.
Function
A relation in which each input value yields a unique output value.
Exactly One Range Value, Or Output
A relation is a set of ordered pairs. A function is a specific type of relation in which each domain value, or input, leads to...
Absolute Value Inequality
A relationship in the form |A| < B or |A| > B.
Joint Variation
A relationship where a variable varies directly or inversely with multiple variables.
Multiplied
A relationship where one quantity is a constant ____ by another quantity is called direct variation.
Divided
A relationship where one quantity is a constant _____ by another quantity is called inverse variation.
Varies Inversely
A relationship where one quantity is a constant divided by the other quantity.
Inversely Proportional
A relationship where one quantity is a constant divided by the other quantity. As one quantity increases, the other decreases.
Varies Directly
A relationship where one quantity is a constant multiplied by the other quantity.
Input
A removable discontinuity might occur in the graph of a rational function if an _____ causes both numerator and denominator to be zero. / Each object or variable in a domain that relates to another object or value by a relationship known as a function. / A function can be shifted vertically by adding a constant to the...
Power Rule For Logarithms
A rule of logarithms that states that the log of a power is equal to the product of the exponent and the log of its base. (logb(M^n)=nlogb(M))
Product Rule For Logarithms
A rule of logarithms that states that the log of a product is equal to a sum of logarithms. (logb(MN)=logb(M)+logb(N))
Quotient Rule For Logarithms
A rule of logarithms that states that the log of a quotient is equal to a difference of logarithms. (logb(M/N)=logb(M)−logb(N))
Descartes' Rule of Signs
A rule that determines the maximum possible numbers of positive and negative real zeros based on the number of sign changes of f(x) and f(−x).
Relation
A set of ordered pairs.
System of Linear Equations
A set of two or more equations in two or more variables that must be considered simultaneously.
Synthetic Division
A shortcut method that can be used to divide a polynomial by a binomial of the form x−k. / Polynomial equations model many real-world scenarios. Solving the equations is easiest done by...
Removable Discontinuity
A single point at which a function is undefined that, if filled in, would make the function continuous. It appears as a hole on the graph of a function.
Extraneous Solution
A solution introduced while solving an equation that does not satisfy the conditions of the original equation.
Consistent System
A system for which there is a single solution to all equations in the system and it is an independent system, or if there are an infinite number of solutions and it is a dependent system.
System of Nonlinear Equations
A system of equations containing at least one equation that is of degree larger than one.
Identity
A system of equations in three variables is dependent if it has an infinite number of solutions. After performing elimination operations, the result is an...
Contradiction
A system of equations in three variables is inconsistent if no solution exists. After performing elimination operations, the result is a...
Simultaneously
A system of linear equations consists of two or more equations made up of two or more variables such that all equations in the system are considered...
Dependent System
A system of linear equations in which the two equations represent the same line. There are an infinite number of solutions to a dependent system.
A Graph
A system of linear equations may also be solved by finding the point of intersection on...
X Value
A system of linear equations may be solved setting the two equations equal to one another and solving for x. The y value may be found by evaluating either one of the original equations using this...
Independent System
A system of linear equations with exactly one solution pair, (x,y).
Inconsistent System
A system of linear equations with no common solution because they represent parallel lines, which have no point or line in common.
Adding a Nonzero Multiple of One Equation to Another Equation
A system of three equations in three variables can be solved by using a series of steps that forces a variable to be eliminated. The steps include interchanging the order of equations, multiplying both sides of an equation by a nonzero constant, and...
System of Nonlinear Inequalities
A system of two or more inequalities in two or more variables containing at least one inequality that is not linear.
Corresponding Variables
A third method of solving a system of linear equations is by addition, in which we can eliminate a variable by adding opposite coefficients of...
Horizontal Compression
A transformation that compresses a function's graph horizontally, by multiplying the input by a constant b>1. (g(x)=f(bx)(b>1)).
Vertical Reflection
A transformation that reflects a function's graph across the x axis by multiplying the output by −1. (g(x)=-f(x)).
Horizontal Reflection
A transformation that reflects a function's graph across the y axis by multiplying the input by -1. (g(x)=f(-x)).
Vertical Shift
A transformation that shifts a function's graph up or down by adding a positive or negative constant to the output. (g(x)=f(x)+k (up for k>0)).
Horizontal Stretch
A transformation that stretches a function's graph horizontally by multiplying the input by a constant 0<b<1. (g(x)=f(bx)(0<b<1)).
Vertical Stretch
A transformation that stretches a function's graph vertically by multiplying the output by a constant a>1. (g(x)=af(x)(a>0)).
Product to Sum Formula
A trigonometric identity that allows the writing of a product of trigonometric functions as a sum or difference of trigonometric functions. (cos α cos β=(1/2)[cos(α−β)+cos(α+β)], sin α cos β=(1/2)[sin(α+β)+sin(α−β)], sin α sin β=(1/2)[cos(α−β)−cos(α+β)], cos α sin β=(1/2)[sin(α+β)−sin(α−β)])
Sum to Product Formula
A trigonometric identity that allows, by using substitution, the writing of a sum of trigonometric functions as a product of trigonometric functions. (sin α+sin β=2 sin((α+β)/2)cos((α−β)/2), sin α−sin β=2 sin((α−β)/2)cos((α+β)/2), cos α−cos β=−2 sin((α+β)/2)sin((α−β)/2), cos α+cos β=2 cos((α+β)/2)cos((α−β)/2))
Degree
A unit of measure describing the size of an angle as one-360th of a full revolution of a circle. / The highest power of the variable that occurs in a polynomial.
Local Minimum
A value of the input where a function changes from decreasing to increasing as the input value increases.
Local Maximum
A value of the input where a function changes from increasing to decreasing as the input value increases.
Correlation Coefficient
A value, r, between -1 and 1 that indicates the degree of linear correlation of variables, or how closely a regression line fits a data set.
Axis of Symmetry
A vertical line drawn through the vertex of a parabola around which the parabola is symmetric; it is defined by x=−b / 2a.
Vertical Asymptote
A vertical line x=a where the graph tends toward positive or negative infinity as the inputs approach a.
X Axis
A vertical reflection reflects a graph about the _____. A graph can be reflected vertically by multiplying the output by -1. / The x intercept is the point at which the graph of a linear function crosses the...
Arrow Notation
A way to symbolically represent the local and end behavior of a function by using arrows to indicate that an input or output approaches a value.
Continuous Growth Formula
A(t)=ae^rt, where t is the number of unit time periods of growth, a is the starting amount, and e is the mathematical constant, e≈2.718282.
Graphically
Absolute value inequalities can also be solved...
Complex Zero
According to the Fundamental Theorem, every polynomial function has at least one...
Eliminate
After solving an exponential equation, check each solution in the original equation to find and _____ any extraneous solutions.
Y Intercept / Initial Value
The value of a function when the input value is 0.
Compounding Periods
The value of an account at any time, t, can be calculated using the compound interest formula when the principal, annual interest rate, and _____ are known.
Mathematical Analysis
The values of trigonometric functions of special angles can be found by...
Quadratic Function
The vertex can be found from an equation representing a... / f(x)=x^2
Not Zero
The vertical asymptotes of a rational function will occur where the denominator of the function is equal to zero and the numerator is...
Amplitude
The vertical height of a function; the constant A appearing in the definition of a sinusoidal function.
Cosine Function
The x-value of the point on a unit circle corresponding to a given angle. (cos t = x)
Sine Function
The y-value of the point on a unit circle corresponding to a given angle. (sin t = y) / The secant, cotangent, and cosecant are all reciprocals of other functions. The secant is the reciprocal of the cosine function, the cotangent is the reciprocal of the tangent function, and the cosecant is the reciprocal of the...
Nominal Rate
The yearly interest rate earned by an investment account, also called annual percentage rate.
Annual Percentage Rate
The yearly interest rate earned by an investment account, also called nominal rate.
Four Solutions, the Circle & the Ellipse Intersect in Four Points
There are five possible types of solutions to the system of nonlinear equations representing an ellipse and a circle: no solution, the circle and the ellipse do not intersect; one solution, the circle and the ellipse are tangent to each other; two solutions, the circle and the ellipse intersect in two points; three solutions, the circle and ellipse intersect in three places...
Simplify a Problem
There are multiple ways to represent a trigonometric expression. Verifying the identities illustrates how expressions can be rewritten to...
One Solution, the Line is Tangent to the Parabola
There are three possible types of solutions to a system of equations representing a circle and a line: no solution, the line does not intersect the circle; _____; two solutions, the line intersects the circle in two points.
Two Solutions, the Line Intersects the Parabola in Two Points.
There are three possible types of solutions to a system of equations representing a line and a parabola: no solution, the line does not intersect the parabola; one solution, the line is tangent to the parabola; and...
Represents
To draw an angle in standard position, draw the initial side along the positive x-axis and then place the terminal side according to the fraction of a full rotation the angle...
Domain
To find the _____ of a logarithmic function, set up an inequality showing the argument greater than zero, and solve for x. / The set of all possible input values for a relation. / The _____ of a composite function consists of those inputs in the _____ of the inner function that correspond to outputs of the inner function that are in the _____ of the outer function.
Zero
To find the zeros of a polynomial function, if it can be factored, factor the function and set each factor equal to... / The domain of a rational function includes all real numbers except those that cause the denominator to equal...
At Most
To graph polynomial functions, find the zeros and their multiplicities, determine the end behavior, and ensure that the final graph has _____ n−1 turning points.
Polynomials
To multiply complex numbers, distribute just as with...
Real, Imaginary
To plot a complex number, we use two number lines, crossed to form the complex plane. The horizontal axis is the _____ axis, and the vertical axis is the _____ axis.
False. A function that is not one to one over its entire domain may be one to one on part of its domain.
True or False: A function that is not one to one over its entire domain may not be one to one on part of its domain.
True
True or False: All translations of the exponential function can be summarized by the general equation f(x)=ab^(x+c)+d.
False. f(x)=alogb(x+c)+d
True or False: All translations of the logarithmic function can be summarized by the general equation f(x)=alogb(x+c)-d.
True
True or False: Each of the toolkit functions has an inverse.
True
True or False: Some quadratic equations must be solved by using the quadratic formula.
True
True or False: Systems of three equations in three variables are useful for solving many different types of real-world problems.
True
True or False: The end behavior of a polynomial function is the same as the end behavior of the power function represented by the leading term of the function.
True
True or False: The equation for a linear function can be written if the slope, m, and initial value, b, are known.
True
True or False: Using the general equation f(x)=alogb(x+c)+d, we can write the equation of a logarithmic function given its graph.
True
True or False: When common logarithms cannot be evaluated mentally, a calculator can be used.
True
True or False: When we are given an exponential equation where the bases are not explicitly shown as being equal, rewrite each side of the equation as powers of the same base, then set the exponents equal to one another and solve for the unknown.
Same Terminal Side
Two angles that have the _____ are called coterminal angles.
Perpendicular Lines
Two lines that intersect at right angles and have slopes that are negative reciprocals of each other.
Parallel Lines
Two or more lines with the same slope.
Change of Base
We can convert a logarithm with any base to a quotient of logarithms with any other base using the ______ formula.
Right Triangle
We can define trigonometric functions as ratios of the side lengths of a...
Side Lengths
We can evaluate the trigonometric functions of special angles, knowing the _____ of the triangles in which they occur.
2π
We can find coterminal angles by adding or subtracting 360° or... / Periodic functions repeat after a given value. The smallest such value is the period. The basic sine and cosine functions have a period of...
t
We can find the age, t, of an organic artifact by measuring the amount, k, of carbon 14 remaining in the artifact and using the formula t=(ln(k))/−0.000121 to solve for...
e
We can solve exponential equations with base ______, by applying the natural logarithm of both sides because exponential and logarithmic functions are inverses of each other.
A Given Time
We can use Newton's Law of Cooling to find how long it will take for a cooling object to reach a desired temperature, or to find what temperature an object will be after...
Local Behavior & End Behavior
We can use arrow notation to describe _____ of the toolkit functions f(x)=1/x and f(x)=1/x^2.
Spread of Rumors
We can use logistic growth functions to model real world situations where the rate of growth changes over time, such as population growth, spread of disease, and...
Best Fit Our Data
We can use real world data gathered over time to observe trends. Knowledge of linear, exponential, logarithmic, and logistic graphs help us to develop models that...
Sum
We can use the product rule of logarithms to rewrite the log of a product as a _____ of logarithms.
Combine or Expand
We can use the product rule, the quotient rule, and the power rule together to _____ a logarithm with a complex input.
Sines & Cosines
We can use the product-to-sum formulas to rewrite products of sines, products of cosines, and products of sine and cosine as sums or differences of...
Difference
We can use the quotient rule of logarithms to rewrite the log of a quotient as a _____ of logarithms.
Any Type of Function
We can use the same problem solving strategies that we would for...
Products
We can use the sum-to-product formulas to rewrite sum or difference of sines, cosines, or products sine and cosine as _____ of sines and cosines.
Unknown
We can use trigonometric functions of an angle to find _____ side lengths. / We can solve many exponential equations by using the rules of exponents to rewrite each side as a power with the same base. Then, we use the fact that exponential functions are one to one to set the exponents equal to one another and solve for the...
logb(S)=logb(T)
When given an equation of the form _____, where S and T are algebraic expressions, we can use the one-to-one property of logarithms to solve the equation S=T for the unknown.
Clockwise
An angle is in standard position if its vertex is at the origin and its initial side lies along the positive x-axis. A positive angle is measured counterclockwise from the initial side and a negative angle is measured...
Quadrantal Angle
An angle whose terminal side lies on an axis.
Horizontal Axis
An angle's reference angle is the size angle, t, formed by the terminal side of the angle t and the...
Endpoints
An average rate of change can also be computed by determining the function values at the _____ of an interval described by a formula.
Variable Exponent
An exponential function is defined as a function with a positive constant other than 1 raised to a...
Data Points
An exponential model can be found using two _____ from the graph of the model.
Calculator
An exponential model can be found using two data points from the graph and a...
Initial Value
An exponential model can be found when the growth rate and _____ are known. / An equation in the slope intercept form of a line includes the slope and the _____ of the function.
Positive Slope
An increasing linear function results in a graph that slants upward from left to right and has a...
Nonlinear Inequality
An inequality containing a nonlinear expression.
The Solution Set
An inequality is graphed in much the same way as an equation, except for > or <, we draw a dashed line and shade the region containing...
Independent Variable
An input variable.
Original Function
An inverse function is one that "undoes" another function. The domain of an inverse function is the range of the original function and the range of an inverse function is the domain of the...
Linear & Angular Speed
An object moving in a circular path has both...
Dependent Variable
An output variable.
Related Functions
An understanding of toolkit functions can be used to find the domain and range of...
Constant
Analyzing the slope within the context of a problem indicates whether a linear function is increasing, decreasing, or...
Second Equation
Another method of solving a system of linear equations is by substitution. In this method, we solve for one variable in one equation and substitute the result into the...
Arccosine
Another name for the inverse cosine; arccos x=(cos^−1)x.
Arcsine
Another name for the inverse sine; arcsin x=(sin^−1)x.
Arctangent
Another name for the inverse tangent; arctan x=(tan^−1)x.
Vertex Form of a Quadratic Function
Another name for the standard form of a quadratic function.
logb(S)=c
When given an equation of the form _____, where S is an algebraic expression, we can use the definition of a logarithm to rewrite the equation as the equivalent exponential equation b^c=S, and solve for the unknown.
Trigonometric Functions
Even and odd properties can be used to evaluate... / Real-world scenarios can be solved using graphs of... / The sum and difference formulas for sine and cosine can also be used for inverse...
Symmetric
Even functions are _____ about the y axis, whereas odd functions are _____ about the origin.
0
Every polynomial function with degree greater than _____ has at least one complex zero. / The x intercept may be found by setting y=0, which is setting the expression mx+b equal to...
Equivalent Logarithmic Form
Exponential equations can be written in their _____ using the definition of a logarithm.
Slope & Y Intercept
When modeling and solving a problem, identify the variables and look for key values, including the...
f(x)=logb(−x)
When the parent function y=logb(x) is multiplied by −1, the result is a reflection about the x axis. When the input is multiplied by −1, the result is a reflection about the y axis. The equation f(x)=−logb(x) represents a reflection of the parent function about the x axis. The equation _____ represents a reflection of the parent function about the y axis. A graphing calculator may be used to approximate solutions to some logarithmic equations
Special Angles
When the sine or cosine is known, we can use the Pythagorean Identity to find the other. The Pythagorean Identity is also useful for determining the sines and cosines of...
Equal
When we are given an exponential equation where the bases are explicitly shown as being _____, set the exponents equal to one another and solve for the unknown.
Cofunction Identities
cos t = sin(3.14/2 - t), sin t = cos(3.14/2 - t), tan t = cot(3.14/2 - t), cot t = tan(3.14/2 - t), sec t = csc(3.14/2 - t), csc t = sec(3.14/2 - t). / sin θ=cos(π/2−θ), cos θ=sin(π/2−θ), tan θ=cot(π/2−θ), cot θ=tan(π/2−θ), sec θ=csc(π/2−θ), csc θ=sec(π/2−θ)
Stretch/Compression
f(x)=Acot(Bx−C)+D is a cotangent with vertical and/or horizontal _____ and shift.
Shift
f(x)=Atan(Bx−C)+D is a tangent with vertical and/or horizontal stretch/compression and...
General Form for the Translation of the Parent Function f(x)=b^x
f(x)=ab^(x+c)+d.
General Form For the Translation of the Parent Logarithmic Function f(x)=logb(x)
f(x)=alogb(x+c)+d
Exponential Function
f(x)=b^x, where b>0, b does not equal 1.
Identity Function
f(x)=x
Cubic Function
f(x)=x^3
Absolute Value Function
f(x)=|x| / An equation of the form |A| = B, with B > 0. It will have solutions when A = B or A = -B.
Square Root Function
f(x)=√x
Difference Formula for Sine
sin(α−β)=sin α cos β−cos α sin β
Carbon 14 Dating
t=(ln(A/A0))/−0.000121. A0 is the amount of carbon 14 when the plant or animal died. A is the amount of carbon 14 remaining today. t is the age of the fossil in years.
Sum Formula for Tangent
tan(α+β)=(tan α+tan β)/(1−tan α tan β)
Difference Formula for Tangent
tan(α−β)=(tan α−tan β)/(1+tan α tan β)
Linear Speed Related to Angular Speed
v=rw.
(sqrt(2)/2, sqrt(2)/2)
π/4
(sqrt(3)/2, 1/2)
π/6
1 Unit
Finding the function values for the sine and cosine begins with drawing a unit circle, which is centered at the origin and has a radius of...
One to One
For a function to have an inverse, it must be...
Definition of a Logarithm
For any algebraic expression, S and positive real numbers b and c, where b does not equal 1, logb(S)=c, if and only if b^c=S.
One to One Property For Logarithmic Functions
For any algebraic expressions, S and T and any positive real number b, where b does not equal 1, logb(S)=logb(T), if and only if S=T.
One to One Property For Exponential Functions
For any algebraic expressions, S and T and any positive real number b, where b^S=b^T, if and only if S=T.
|a|>1
For any constant a>0, the equation f(x)=alogb(x) stretches the parent function y=logb(x) vertically by a factor of a if _____ or compresses the parent function y=logb(x) vertically by a factor of a if |a|<1.
Inverse Function
For any one to one function f(x), the inverse is a function f^−1 (x) such that f^−1 (f(x))=x for all x in the domain of f. This also implies that f(f^−1 (x))=x for all x in the domain of f^−1.
f
For any trigonometric function f(x), if x=f^−1(y), then f(x)=y. However, f(x)=y only implies x=f^−1(y) if x is in the restricted domain of...
Logarithmic Function
For x>0, b>0, b≠1, y=logb(x) if and only if b^y=x.
Input, Output
Function notation is a shorthand method for relating the _____ to the _____ in the form y=f(x).
More Than One Way
Functions can often be decomposed in...
New Identities
Fundamental identities such as the Pythagorean Identity can be manipulated algebraically to produce...
x=-c
Given an equation with the general form f(x)=alogb(x+c)+d, we can identify the vertical asymptote _____ for the transformation.
Evaluate
Identities can be used to _____ trigonometric functions. / To _____ a function, we determine an output value for a corresponding input value. Algebraic forms of a function can be evaluated by replacing the input variable with a given value.
Reduction Formulas
Identities derived from the double-angle formulas and used to reduce the power of a trigonometric function. ((sin^2)θ=(1−cos(2θ))/2, (cos^2)θ=(1+cos(2θ))/2, (tan^2)θ=(1−cos(2θ))/(1+cos(2θ)))
Half Angle Formulas
Identities derived from the reduction formulas and used to determine half-angle values of trigonometric functions. (sin (α/2)=±sqrt((1−cos α)/2), cos (α/2)=±sqrt((1+cos α)/2), tan (α/2)=±sqrt((1−cos α)/(1+cos α)=sin α/(1+cos α)=(1−cos α)/sin α)
Double Angle Formulas
Identities derived from the sum formulas for sine, cosine, and tangent in which the angles are equal. (sin(2θ)=2sin θ cos θ, cos(2θ)=(cos^2)θ−(sin^2)θ=1−2(sin^2)θ=2(cos^2)θ−1, tan(2θ)=2tan θ/(1−(tan^2)θ)
0<b<1
If _____, the function is decreasing. The left tail of the graph will increase without bound, and the right tail will approach the asymptote y=0.
b>1
If _____, the function is increasing. The left tail of the graph will approach the asymptote y=0, and the right tail will increase without bound.
Remainder Theorem
If a polynomial f(x) is divided by x−k, then the remainder is equal to the value f(k).
x=x1,x2,...,xn
If a rational function has x-intercepts at _____, vertical asymptotes at x=v1,v2,...,vm, and no xi=any vj, then the function can be written in the form f(x)=a((x−x1)^p1 (x−x2)^p2⋯(x−xn)^pn/(x−v1)^q1 (x−v2)^q2⋯(x−vm)^qn)
g(f(x))=f(g(x))=x
If g(x) is the inverse of f(x), then...
Two Solutions
If the absolute value of an equation is set equal to a positive number, expect _____ for the unknown variable.
−π/2≤x≤π/2
If the inside function is a trigonometric function, then the only possible combinations are sin^−1(cos x)=(π/2)−x if 0≤x≤π and cos^−1(sin x)=(π/2)−x if...
Tabular
In _____ form, a function can be represented by rows or columns that relate to input and output values. / A function presented in _____ form can also be reflected by multiplying the values in the input and output rows or columns accordingly.
Carrying Capacity
In a logistic model, the limiting value of the output.
Adjacent Side
In a right triangle, the side between a given angle and the right angle.
Opposite Side
In a right triangle, the side most distant from a given angle.
Radians
In addition to degrees, the measure of an angle can be described in...
Unknown Variable
In an absolute value equation, an _____ is the input of an absolute value function.
Extrema
Minima and maxima are also called...
Distances
Right-triangle trigonometry permits the measurement of inaccessible heights and... / The absolute value function is commonly used to measure _____ between points.
[exp(x)]
Scientific and graphing calculators have the key [e^x] or ______ for calculating powers of e.
Known Side
Select the trigonometric function representing the ratio of the unknown side to the...
Reciprocal Identities
Set of equations involving the reciprocals of basic trigonometric definitions. (sin θ=1/csc θ, cos θ=1/sec θ, tan θ=1/cot θ, csc θ=1/sin θ, sec θ=1/cos θ, cot θ=1/tan θ)
Pythagorean Identities
Set of equations involving trigonometric functions based on the right triangle properties. (sin^2θ+cos^2θ=1, 1+cot^2θ=csc^2θ, 1+tan^2θ=sec^2θ)
Even Odd Identities
Set of equations involving trigonometric functions such that if f(−x)=−f(x), the identity is odd, and if f(−x)=f(x), the identity is even. (tan(−θ)=−tan θ, cot(−θ)=−cot θ, sin(−θ)=−sin θ, csc(−θ)=−csc θ, cos(−θ)=cos θ, sec(−θ)=sec θ)
π/6=sin^−1(1/2)
Special angles are the outputs of inverse trigonometric functions for special input values; for example, π/4=tan^−1(1) and...
Verifying
Sum and difference formulas are useful in _____ identities.
x-k
Synthetic division is a shortcut that can be used to divide a polynomial by a binomial in the form...
No Solution
Systems of equations are classified as independent with one solution, dependent with an infinite number of solutions, or inconsistent with...
Profit
Systems of equations can be used to solve real-world problems that involve more than one variable, such as those relating to revenue, cost, and...
Two Identical Planes That Intersect the Third On a Line
Systems of equations in three variables that are dependent could result from three identical planes, three planes intersecting at a line, or...
Not At the Same Location
Systems of equations in three variables that are inconsistent could result from three parallel planes, two parallel planes and one intersecting plane, or three planes that intersect the other two but...
Remainder
The Division Algorithm tells us that a polynomial dividend can be written as the product of the divisor and the quotient added to the...
f(c)=0
The Intermediate Value Theorem tells us that if f(a) and f(b) have opposite signs, then there exists at least one value c between a and b for which _____.
Cofunction
The _____ identities apply to complementary angles and pairs of reciprocal functions.
Domain, Range
The _____ of a quadratic function is all real numbers. The _____ varies with the function.
Inverse
The _____ of an exponential function is a logarithmic function, and the _____ of a logarithmic function is an exponential function. / To find the _____ of a formula, solve the equation y=f(x) for x as a function of y. Then, exchange the labels x and y.
Linear Speed of An Object
The _____ traveling along a circular path is the distance it travels in a unit of time.
Angle of Elevation
The angle between the horizontal and the line from the object to the observer's eye, assuming the object is positioned higher than the observer.
Angle of Depression
The angle between the horizontal and the line from the object to the observer's eye, assuming the object is positioned lower than the observer.
Angular Speed
The angle through which a rotating object travels in a unit of time. (w=0/t)
Fraction of the Area
The area of sector is a _____ of the entire circle.
Expression
The average rate of change can sometimes be determined as an...
Vertex
The common endpoint of two rays that form an angle. / The point at which a parabola changes direction, corresponding to the minimum or maximum value of the quadratic function.
Complex Conjugate
The complex number in which the sign of the imaginary part is changed and the real part of the number is left unchanged; when added to or multiplied by the original complex number, the result is a real number.
Degree of Linear Relationship
The correlation coefficient, r, indicates the _____ between data.
±2π...
The cotangent function has period π and vertical asymptotes at 0,±π,...
A/x+(Bx+C)/(ax^2+bx+c)
The decomposition of (P(x))/(Q(x)) with a nonrepeated irreducible quadratic factor needs a linear numerator over the quadratic factor, as in...
Increasing Powers
The decomposition of (P(x))/(Q(x)) with repeated linear factors must account for the factors of the denominator in .
Highest
The degree of a polynomial function is the _____ power of the variable that occurs in a polynomial. The term containing the highest power of the variable is called the leading term. The coefficient of the leading term is called the leading coefficient.
Average Rate of Change
The difference in the output values of a function found for two values of the input divided by the difference between the inputs. (◇y / ◇x = f(x2)-f(x1) / x2-x1)
Linear Speed
The distance along a straight path a rotating object travels in a unit of time; determined by the arc length. (v=s/t)
Ordered Pairs
The domain of a function can be determined by listing the input values of a set of...
Negative Number
The domain of a function includes all real input values that would not cause us to attempt an undefined mathematical operation, such as dividing by zero or taking the square root of a...
All Real Numbers
The domain of the sine and cosine functions is...
f(x)=b^(x+c)
The equation _____ represents a horizontal shift of the parent function f(x)=b^x.
f(x)=b^x+d
The equation _____ represents a vertical shift of the parent function f(x)=b^x.
a>0
The equation f(x)=ab^x, where _____, represents a vertical stretch if |a|>1 or compression if 0<|a|<1 of the parent function f(x)=b^x.
d<0
The equation f(x)=logb(x)+d shifts the parent function y=logb(x) vertically up d units if d>0 or down d units if...
y=logb(x)
The equation f(x)=logb(x+c) shifts the parent function _____ horizontally left c units if c>0 or right c units if c<0.
Slope Intercept Form
The equation for a line that represents a linear function in the form f(x)=mx+b.
Point Slope Form
The equation for a line that represents a linear function of the form y-y1=m(x-x1). / A line parallel to another line, passing through a given point, may be found by substituting the slope value of the line and the x and y values of the given point into the equation f(x)=mx+b and using the b that results. Similarly, the _____ of an equation can be used.
Graph
The equation for a sinusoidal function can be determined from a... / For many functions, the domain and range can be determined from a... / We can find local extrema from a... / A composite function can be evaluated from a...
Common Logarithm
The exponent to which 10 must be raised to get x; log10(x) is written simply as log(x). (For x>0, y=log(x) if and only if 10^y=x.)
Logarithm
The exponent to which b must be raised to get x; written y=logb(x).
Natural Logarithm
The exponent to which the number e must be raised to get x; loge(x) is written as ln(x). (For x>0, y=ln(x) if and only if e^y=x.)
Combining
The function produced by _____ two functions is a composite function.
Y Axis
The function sin x is odd, so its graph is symmetric about the origin. The function cos x is even, so its graph is symmetric about the... / The axis of symmetry is the vertical line passing through the vertex. The zeros, or x intercepts, are the points at which the parabola crosses the x axis. The y intercept is the point at which the parabola crosses the... / When the parent function f(x)=b^x is multiplied by −1, the result, f(x)=−b^x, is a reflection about the x axis. When the input is multiplied by −1, the result, f(x)=b^−x, is a reflection about the... / A horizontal reflection reflects a graph about the _____. A graph can be reflected horizontally by multiplying the input by -1. / The initial value, or y intercept, is the output value when the input of a linear function is 0. It is the y value of the point at which the line crosses the...
Standard Form of a Quadratic Function
The function that describes a parabola, written in the form f(x)=a(x−h)^2+k, where (h, k) is the vertex.
General Form of a Quadratic Function
The function that describes a parabola, written in the form f(x)=ax^2+bx+c, where a,b, and c are real numbers and a≠0.
Revenue Function
The function that is used to calculate revenue, simply written as R=xp, where x=quantity and p=price.
Cost Function
The function used to calculate the costs of doing business. It usually has two parts, fixed costs and variable costs.
Horizontal Line
The graph of a one to one function passes the _____ test. / A constant linear function results in a graph that is a... / A line defined by f(x)=b, where b is a real number. The slope of a horizontal line is 0.
Turning Points
The graph of a polynomial function changes direction at its...
Even
The graph of a polynomial will touch the horizontal axis at a zero with _____ multiplicity.
Either Up or Down
The graph of a quadratic function is a parabola. A parabola is a U-shaped curve that can open...
Sine or Cosine Function
The graph of a sinusoidal function has the same general shape as a...
y=x
The graph of an inverse function is the reflection of the graph of the original function across the line...
Corner Point
The graph of the absolute value function resembles a letter V. It has a _____ at which the graph changes direction.
y=0
The graph of the function f(x)=b^x has a y-intercept at (0, 1), domain (−∞, ∞), range (0, ∞), and horizontal asymptote...
x=0
The graph of the parent function f(x)=logb(x) has an x-intercept at (1,0), domain (0,∞), range (−∞,∞), vertical asymptote _____, and, if b>1, the function is increasing, or if 0<b<1, the function is decreasing.
Absolute Maximum
The greatest value of a function over an interval.
Phase Shift
The horizontal displacement of the basic sine or cosine function; the constant C/B.
Midline
The horizontal line y=D, where D appears in the general form of a sinusoidal function.
Other Side
The identities can be verified using other formulas or by converting the expressions to sines and cosines. To verify an identity, we choose the more complicated side of the equals sign and rewrite it until it is transformed into the...
Partial Fractions
The individual fractions that make up the sum or difference of a rational expression before combining them into a simplified rational expression.
Life Span
The initial investment of an account can be found using the compound interest formula when the value of the account, annual interest rate, compounding periods, and _____ of the account are known.
Minimizing
The least squares regression line is found by ____ the squares of the distances of points from a line passing through the data and may be used to make predictions regarding either of the variables.
Entire Circle
The length of a circular arc is a fraction of the circumference of the...
Arc Length
The length of the curve formed by an arc. (s=r0)
Half Life
The length of time it takes for a substance to exponentially decay to half of its original quantity. (If A=A0e^(kt), k<0, the half life is t=−(ln(2)/k).)
Best Fit
The line of _____ may be estimated or calculated, using a calculator or statistical software.
Turning Point
The location at which the graph of a function changes direction.
Absolute Minimum
The lowest value of a function over an interval.
Radian
The measure of a central angle of a circle that intercepts an arc equal in length to the radius of that circle.
Reference Angle
The measure of the acute angle formed by the terminal side of the angle and the horizontal axis. / The sine and cosine of an angle have the same absolute value as the sine and cosine of its...
Revenue
The minimum or maximum value of a quadratic function can be used to determine the range of the function and to solve many kinds of real-world problems, including problems involving area and revenue.
Behaves At the X Intercepts
The multiplicity of a zero determines how the graph...
Composite Function
The new function formed by function composition, when the output of one function is used as the input of another. ((f ○ g)(x) = f(g(x)))
Constant of Variation
The non zero value k that helps define the relationship between variables in direct or inverse variation.
2.718282
The number e is a mathematical constant often used as the base of real world exponential growth and decay models. Its decimal approximation is e≈...
Less Than
The number of negative real zeros of a polynomial function is either the number of sign changes of f(−x) or _____ the number of sign changes by an even integer.
Even Integer
The number of positive real zeros of a polynomial function is either the number of sign changes of the function or less than the number of sign changes by an...
Multiplicity
The number of times a given factor appears in the factored form of the equation of a polynomial. If a polynomial contains a factor of the form (x−h)^p, x=h is a zero of multiplicity p.
The Order Given
The order in which different transformations are applied does affect the final function. Both vertical and horizontal transformations must be applied in _____. However, a vertical transformation may be combined with a horizontal transformation in any order.
The Meaning of Compisite Functions
The order of function composition must be considered when interpreting...
Points On a Line
The ordered pairs given by a linear function represent...
x
The period P of a repeating function f is the smallest interval such that f(x+P)=f(x) for any value of...
Range
The range of cotangent is (−∞,∞), and the function is decreasing at each point in its... / The set of output values that result from the input values in a relation.
Radian Measure
The ratio of the arc length formed by an angle divided by the radius of the circle.
Slope
The ratio of the change in output values to the change in input values. A measure of the steepness of a line. (m=change in output / change in input = ◇y / ◇x = y2-y1 / x2-x1) / Parallel lines have the same...
Secant
The reciprocal of the cosine function: on the unit circle, sec t=1/x,x≠0. (sec t = 1 / cos t)
Cosecant
The reciprocal of the sine function: on the unit circle, csc t=1/y,y≠0. (csc t = 1 / sin t) / f(x)=Acsc(Bx−C)+D gives a shifted, compressed, and/or stretched _____ function graph.
Inverse Variation
The relationship between two variables in which the product of the variables is a constant. (y=k/(x^n), k is a nonzero constant)
Direct Variation
The relationship between two variables that are a constant multiple of each other; as one quantity increases, so does the other. (y=kx^n, k is a nonzero constant)
Acute Angle
The same side lengths can be used to evaluate the trigonometric functions of either _____ in a right triangle.
Newton's Law of Cooling
The scientific formula for temperature as a function of time as an object's temperature is equalized with the ambient temperature. (T(t)=Ae^(kt)+Ts, where Ts is the ambient temperature, A=T(0)−Ts, and k is the continuous rate of cooling.
Function Graph
The secant and cosecant are both periodic functions with a period of 2π. f(x)=Asec(Bx−C)+D gives a shifted, compressed, and/or stretched secant...
Solution Set
The set of all ordered pairs or triples that satisfy all equations in a system of equations.
Hypotenuse
The side of a right triangle opposite the right angle.
Terminal Side
The side of an angle at which rotation ends.
Initial Side
The side of an angle from which rotation begins.
Original Angle
The signs of the sine and cosine are determined from the x- and y-values in the quadrant of the...
Period
The smallest interval P of a repeating function f such that f(x+P)=f(x). / A function can be graphed by identifying its amplitude and...
The Same Line
The solution to a system of dependent equations will always be true because both equations describe...
Independently
The solution to a system of linear equations in two variables is any ordered pair that satisfies each equation...
Feasible Region
The solution to a system of nonlinear inequalities that is the region of the graph where the shaded regions of each inequality intersect.
Multiple
The square root of any negative number can be written as a _____ of i.
Second Angle
The sum formula for sines states that the sine of the sum of two angles equals the product of the sine of the first angle and cosine of the second angle plus the product of the cosine of the first angle and the sine of the second angle. The difference formula for sines states that the sine of the difference of two angles equals the product of the sine of the first angle and cosine of the second angle minus the product of the cosine of the first angle and the sine of the...
Product
The sum formula for tangent states that the tangent of the sum of two angles equals the sum of the tangents of the angles divided by 1 minus the product of the tangents of the angles. The difference formula for tangent states that the tangent of the difference of two angles equals the difference of the tangents of the angles divided by 1 plus the _____ of the tangents of the angles. / We can use the power rule for logarithms to rewrite the log of a power as the ______ of the exponent and the log of its base.
Complex Number
The sum of a real number and an imaginary number, written in the standard form a+bi, where a is the real part, and bi is the imaginary part. / Allowing for multiplicities, a polynomial function will have the same number of factors as its degree. Each factor will be in the form (x−c), where c is a...
π
The tangent function has period...
Fundamental Identities
Verifying an identity may involve algebra with the...
Combined
Vertical and horizontal shifts are often...
Can
We _____ perform algebraic operations on functions.
Substitution
We can also derive the sum-to-product identities from the product-to-sum identities using...
y=c
We can also use graphing to solve equations with the form logb(S)=c. We graph both equations y=logb(S) and _____ on the same coordinate plane and identify the solution as the x value of the intersecting point.
k>0
We can also write this formula in terms of continuous growth as A=A0e^(kx), where A0 is the starting value. If A0 is positive, then we have exponential growth when k>0 and exponential decay when...
Simplifying
We can create an identity by _____ an expression and then verifying it.
General Form
Quadratic functions are often written in _____. Standard or vertex form is useful to easily identify the vertex of a parabola. Either form can be written from a graph.
Change of Base Formula
A formula for converting a logarithm with any base to a quotient of logarithms with any other base. (logb(M)=(logn(M)/logn(b)), n>0, n does not equal 1, b does not equal 1)
Complementary
Any two _____ angles could be the two acute angles of a right triangle.
Rates & Concentrations
Application problems involving _____ often involve rational functions.
Area of a Sector
Area of a portion of a circle bordered by two radii and the intercepted arc; the fraction θ/2π multiplied by the area of the entire circle. (A=1/20r^2)
A Calculator
Real-world exponential problems with base 10 can be rewritten as a common logarithm and then evaluated using...
Trigonometric Term
Reduction formulas are especially useful in calculus, as they allow us to reduce the power of the...
Intersection
A solution set is an ordered triple {(x,y,z)} that represents the _____ of three planes in space.
(sqrt(3)/2, -1/2)
11π/6
(-1/2, sqrt(3)/2)
2π/3
(-sqrt(2)/2, sqrt(2)/2)
3π/4
(-1/2, -sqrt(3)/2)
4π/3
(sqrt(2)/2, -sqrt(2)/2)
7π/4
Unit Circle
A circle with a center at (0,0) and radius 1. / The tangent of an angle is the ratio of the y-value to the x-value of the corresponding point on the...
Formula
A composite function can be evaluated from a...
Inner Function
A composition function can be evaluated by evaluating the _____ using the given input value and then evaluating the outer function taking as its input the output of the _____.
Set Buider Notation
A method of describing a set by a rule that all of its members obey. It takes the form {x | statement about x}.
Negative Slope
A decreasing linear function results in a graph that slants downward from left to right and has a...
Smooth Curve
A graph with no sharp corners.
Tabular Form
A linear function can be written from...
Exponential Growth
A model that grows by a rate proportional to the amount present. (f(x)=ab^x, where a>0, b>0, b does not equal 1.)
The Data
A regression line best fits...
Least Squares Regression
A statistical technique for fitting a line to data in a way that minimizes the differences between the line and data values.
Fundamental Rules of Mathematics
Algebraic techniques can be used to simplify trigonometric expressions. We use algebraic techniques throughout this text, as they consist of the...
Linear Factorization Theorem
Allowing for multiplicities, a polynomial function will have the same number of factors as its degree, and each factor will be in the form (x−c), where c is a complex number.
X Intercepts
Another way to find the _____ of a polynomial function is to graph the function and identify the points at which the graph crosses the x axis.
No Solutions
An absolute value equation may have one solution, two solutions, or...
Boundaries
An absolute value inequality is similar to an absolute value equation but takes the form |A| < B or |A| > B. It can be solved by determining the boundaries of the solution set and then testing which segments are in the set.
Equation
An algebraic form of a function can be written from an... / The domain of a function can also be determined by identifying the input values of a function written as an... / The y intercept and slope of a line may be used to write the _____ of a line.
Substitution Method
An algebraic technique used to solve systems of linear equations in which one of the two equations is solved for one variable and then substituted into the second equation to solve for the second variable.
Addition Method
An algebraic technique used to solve systems of linear equations in which the equations are added in a way that eliminates one variable, allowing the resulting equation to be solved for the remaining variable. Substitution is then used to solve for the first variable.
Amount of Rotation
An angle is formed from the union of two rays, by keeping the initial side fixed and rotating the terminal side. The _____ determines the measure of the angle.
Term of a Polynomial Function
Any aix^i of a polynomial function in the form f(x)=anx^n+a(n-1)x^(n-1)+...+a2x^2+a1x+a1.
k=lnb
Any exponential function with the form y=abx can be rewritten as an equivalent exponential function with the form y=A0e^(kx) where...
Sinusoidal Function
Any function that can be expressed in the form f(x)=Asin(Bx−C)+D or f(x)=Acos(Bx−C)+D.
Equal or Lower
Polynomial long division can be used to divide a polynomial by any polynomial with _____ degree.
Interpolation
Predicting a value inside the domain and range of the data.
Restricted Function
Because the trigonometric functions are not one-to-one on their natural domains, inverse trigonometric functions are defined for...
Extrapolation
Predicting a value outside the domain and range of the data.
Entering Information
Calculators and graphing software are helpful for finding sines and cosines if the proper procedure for _____ is known.
Reasonableness
Check for _____ of the answer.
Local Extrema
Collectively, all of a function's local maxima and minima.
Detected
Combinations of variations of sinusoidal functions can be _____ from an equation.
Exponent
Combining the skills learned in this and previous sections, we can solve equations that model real world situations, whether the unknown is in an _____ or in the argument of a logarithm.
10
Common logarithms can be evaluated mentally using previous knowledge of powers of...
Imaginary Parts
Complex numbers can be added and subtracted by combining the real parts and combining the...
And
Complex numbers can be multiplied (and/or) divided.
Growth or Decay Rate
Continuous growth or decay models are exponential models that use e as the base. Continuous growth and decay models can be found when the initial value and _____ are known.
Odd
Cosine and secant are even; sine, tangent, cosecant, and cotangent are... / The graph of a polynomial will cross the horizontal axis at a zero with _____ multiplicity.
Degrees
Coterminal angles can be found using radians just as they are for...
System of Equations
Decompose (P(x))/(Q(x)) by writing the partial fractions as A/(a1x+b1)+B/(a2x+b2). Solve by clearing the fractions, expanding the right side, collecting like terms, and setting corresponding coefficients equal to each other, then setting up and solving a...
Negative Angle
Description of an angle measured clockwise from the positive x-axis.
Positive Angle
Description of an angle measured counterclockwise from the positive x-axis.
Coterminal Angles
Description of positive and negative angles in standard position sharing the same terminal side.
Intersect
Either method of solving a system of equations results in a false statement for inconsistent systems because they are made up of parallel lines that never...
Sine, Cosine, & Tangent.
Double-angle identities are derived from the sum formulas of the fundamental trigonometric functions:
Tabular Function
For a _____, exchange the input and output rows to obtain the inverse.
Intermediate Value Theorem
For two numbers a and b in the domain of f, if a<b and f(a)≠f(b), then the function f takes on every value between f(a) and f(b), specifically when a polynomial function changes from a negative value to a positive value, the function must cross the x axis.
Division Algorithm
Given a polynomial dividend f(x) and a non zero polynomial divisor d(x) where the degree of d(x) is less than or equal to the degree of f(x), there exist unique polynomials q(x) and r(x) such that f(x)=d(x)q(x)+r(x) where q(x) is the quotient and r(x) is the remainder. The remainder is either equal to zero or has degree strictly less than d(x). (f(x)=d(x)q(x)+r(x) where q(x)≠0)
Exponential Growth & Decay
Given a substance's doubling time or half time, we can find a function that represents its...
Verify it
Graphing both sides of an identity will...
Transformed
Graphs of linear functions may be _____ by using shifts up, down, left, or right, as well as through stretches, compressions, and reflections.
Known or Not
Half-angle formulas allow us to find the value of trigonometric functions involving half-angles, whether the original angle is...
Global Maximum
Highest turning point on a graph, f(a), where f(a)≥f(x) for all x.
Vice Versa
If two angles are complementary, the cofunction identities state that the sine of one equals the cosine of the other and...
sin(cos^−1(x))=sqrt(1−x^2)
In function composition, if the inside function is an inverse trigonometric function, then there are exact expressions; for example...
Zeros
In a given function, the values of x at which y=0, also called roots.
Growth & Decay
In general, we solve problems involving exponential growth or decay in two steps. First, we set up a model and use the model to find the parameters. Then we use the formula with these parameters to predict...
Multiple Variables
In many problems, a variable varies directly or inversely with _____. We call this type of relationship joint variation.
(Ax+B)/(ax^2+bx+c)+(A2x+B2)/(ax^2+bx+c)^2+⋯+(Anx+Bn)/(ax^2+bx+c)^n
In the decomposition of (P(x))/(Q(x)), where Q(x) has a repeated irreducible quadratic factor, when the irreducible quadratic factors are repeated, powers of the denominator factors must be represented in increasing powers as...
2π/|B|
In the general formula for a sinusoidal function, the period is P=...
Compressed
In the general formula for a sinusoidal function, |A| represents amplitude. If |A|>1, the function is stretched, whereas if |A|<1, the function is...
Both Inequalities
Inequalities are solved the same way as equalities, but solutions to systems of inequalities must satisfy...
Table
Input and output values of a function can be identified from a... / Comparing pairs of input and output values in a _____ can also be used to find the average rate of change. / A composite function can be evaluated from a...
Compound Interest
Interest earned on the total balance, not just the principal. (A(t)=P(1+r/n)^nt, where A(t) is the account value of time t, t is the number of years, P is the initial investment, often called the principal, r is the annual percentage rate, or nomial rate, and n is the number of compounding periods in one year.)
Number Line
Interval values represented on a _____ can be described using inequality notation, set builder notation, and interval notation.
Adding the Two Equations Together
It is often necessary to multiply one or both equations by a constant to facilitate elimination of a variable when...
Simpler Functions
Just as functions can be combined to form a composite function, composite functions can be decomposed into...
Graphical Form
Linear functions can be represented in words, function notation, tabular form, and...
Plotting Points
Linear functions may be graphed by ______ or by using the y intercept and slope.
Identifying, Using
Linear models may be built by _____ or calculating the slope and _____ the y intercept.
Equivalent Exponential Form
Logarithmic equations can be written in an _____, using the definition of a logarithm.
b
Logarithmic functions with base b can be evaluated mentally using previous knowledge of powers of...
Global Minimum
Lowest turning point on a graph, f(a), where f(a)≤f(x) for all x.
Same Set of Axes
One method of solving a system of linear equations in two variables is by graphing. In this method, we graph the equations on the...
Ray
One point on a line and all points extending in one direction from that point; one side of an angle.
Function Notation
One type of _____ is the slope intercept form of an equation.
Quotient Identities
Pair of identities based on the fact that tangent is the ratio of sine and cosine, and cotangent is the ratio of cosine and sine. (tan θ=sin θ/cos θ, cot θ=cos θ/sin θ)
Vertical
Perpendicular lines have negative reciprocal slopes, assuming neither is...
Area & Volume
Polynomial division can be used to solve application problems, including...
Unit of Time
The angular speed of an object traveling in a circular path is the measure of the angle through which it turns in a...
Point On a Circle
Reference angles can also be used to find the coordinates of a...
Sine & Cosine
Reference angles can be used to find the _____ of the original angle. / From the sum and difference identities, we can derive the product-to-sum formulas and the sum-to-product formulas for...
Evaluate a Function
Relating input values to output values on a graph is another way to...
Vertical & Horizontal Shifts
Relating the shift to the context of a problem makes it possible to compare and interpret...
Linear or Nonlinear Models
Scatter plots may represent...
Verifying An Identity
Simplifying one side of the equation to equal the other side is another method for...
Real World Problems
Sinusoidal functions can be used to solve... / A linear function can be used to solve...
Identities
Statements that are true for all values of the input on which they are defined.
Cosine
The Pythagorean Identity makes it possible to find a cosine from a sine or a sine from a...
Slope & Initial Value
The _____ can be determined given a graph or any two points on a line.
Measure of An Angle
The amount of rotation from the initial side to the terminal side.
More Complex Side
The approach to verifying an identity depends on the nature of the identity. It is often useful to begin on the _____ of the equation.
Decay
The basic exponential function is f(x)=ab^x. If b>1, we have exponential growth. If 0<b<1, we have exponential...
Decreases, Increases
The behavior of a graph as the input _____ beyond bound and _____ beyond bound is called the end behavior.
End Behavior
The behavior of the graph of a function as the input decreases without bound and increases without bound. / Graph rational functions by finding the intercepts, behavior at the intercepts and asymptotes, and...
Rate of Change
The change of an output quantity relative to the change of the input quantity. / The _____ of a linear function is also known as the slope.
10 and e
The change of base formula is often used to rewrite a logarithm with a base other than ______ as the quotient of natural or common logs. That way a calculator can be used to evaluate.
Leading Coefficient
The coefficient of the leading term. / Each rational zero of a polynomial function with integer coefficients will be equal to a factor of the constant term divided by a factor of the...
±3π/2
The cotangent is zero at ±π/2...
Even or Odd
The end behavior depends on whether the power is...
Inverse Cosine Function
The function (cos^−1)x, which is the inverse of the cosine function and the angle that has a cosine equal to a given number.
Inverse Sine Function
The function (sin^−1)x, which is the inverse of the sine function and the angle that has a sine equal to a given number.
Inverse Tangent Function
The function (tan^−1)x, which is the inverse of the tangent function and the angle that has a tangent equal to a given number.
At Specific Points On a Graph
The inverse of a function can be determined...
Break Even Point
The point at which a cost function intersects a revenue function. Where profit is zero.
The Same
The point of intersection of a system of linear equations is the point where the x and y values are...
X Intercept
The point on the graph of a linear function when the output is 0. The point at which the graph crosses the horizontal axis.
A Line Passes
The point slope form is also convenient for finding a linear equation when given two points through which...
One Point
The point slope form is useful for finding a linear equation when given the slope of a line and...
Standard Position
The position of an angle having the vertex at the origin and the initial side along the positive x-axis.
Rational Zero Theorem
The possible rational zeros of a polynomial function have the form pq where p is a factor of the constant term and q is a factor of the leading coefficient.
Order of Magnitude
The power of ten, when a number is expressed in scientific notation, with one non-zero digit to the left of the decimal.
Fourth One
The powers of i are cyclic, repeating every...
Partial Fraction Decomposition
The process of returning a simplified rational expression to its original form, a sum or difference of simpler rational expressions.
Profit Function
The profit function is written as P(x)=R(x)-C(x), revenue minus cost.
Tangent
The quotient of the sine and cosine: on the unit circle, tan t=y/x,x≠0. (tan t = sin t / cos t) / The sum and difference formulas can be used to find the exact values of the sine, cosine, or _____ of an angle.
[-1, 1]
The range of both the sine and cosine functions is...
Condense
The rules of logarithms can also be used to _____ sums, differences, and products with the same base as a single logarithm.
Axis
The sine and cosine values are most directly determined when the corresponding point on the unit circle falls on an...
A Point
The six trigonometric functions can be found from ___ on the unit circle.
Any Two Points On the Line
The slope of a linear function can be calculated by dividing the difference between y values by the difference in corresponding x values of...
Angles
The sum formula for cosines states that the cosine of the sum of two angles equals the product of the cosines of the angles minus the product of the sines of the angles. The difference formula for cosines states that the cosine of the difference of two angles equals the product of the cosines of the angles plus the product of the sines of the...
Leading Term
The term containing the highest power of the variable. / The end behavior of a polynomial function depends on the...
Doubling Time
The time it takes for a quantity to double. (If A=A0e^(kt), k>0, the doubling time is t=(ln2)/k.)
Regular Intervals
The trigonometric functions repeat at...
Angle
The union of two rays having a common endpoint. / Trigonometric functions can also be found from an...
Known
The unknown height or distance can be found by creating a right triangle in which the unknown height or distance is one of the sides, and another side and angle are...
General Formula
The value D in the ____ for a sinusoidal function indicates the vertical shift from the midline.
C/B
The value _____ in the general formula for a sinusoidal function indicates the phase shift.
Solve For
To _____ a specific function value, we determine the input values that yield the specific output value.
(θ^R)/π
To convert between degrees and radians, use the proportion θ/180=...
Denominator
To divide complex numbers, multiply both the numerator and denominator by the complex conjugate of the denominator to eliminate the complex number from the...
Computer Software
To evaluate trigonometric functions of other angles, we can use a calculator or...
Formulas
Trigonometric expressions are often simpler to evaluate using the...
Reference Angles
Trigonometric functions of angles outside the first quadrant can be determined using...
Constant Ratio
Two variables that are directly proportional to one another will have a...
Constant Multiple
Two variables that are inversely proportional to one another will have a...
f(x)=ab^(x+c)+d
Using the general equation _____, we can write the equation of a function given its description.
Endpoint
Using the unit circle, the sine of an angle t equals the y-value of the endpoint on the unit circle of an arc of length t whereas the cosine of an angle t equals the x-value of the...
Model Breakdown
When a model no longer applies after a certain point.
Taking the Logarithm of Each Side
When an exponential equation cannot be rewritten with a common base, solve by...
Output
When evaluating the composition of a trigonometric function with an inverse trigonometric function, draw a reference triangle to assist in determining the ratio of sides that represents the _____ of the trigonometric function. / Each object or value in the range that is produced when an input value is entered into a function.
Ratio of Sides
When evaluating the composition of a trigonometric function with an inverse trigonometric function, you may use trig identities to assist in determining the...
First, Second
When functions are combined, the output of the _____ function becomes the input of the _____ function.
Constant Term
When the leading coefficient is 1, the possible rational zeros are the factors of the...
Difference Formula For Cosine
cos(α−β)=cos α cos β+sin α sin β
Reciprocal Squared Function
f(x)=1 / x^2
Cube Root Function
f(x)=3√x
Constant Function
f(x)=c, where c is a constant
Factor Theorem
k is a zero of polynomial function f(x) if and only if (x−k) is a factor of f(x).
Cotangent
the reciprocal of the tangent function: on the unit circle, cot t=x/y,y≠0. (cot t = 1 / tan t = cos t / sin t)
Shifted, Compressed, And/Or Stretched Cosecant Function
y=Acsc(Bx-C)+D
Shifted, Compressed, And/Or Stretched Secant Function
y=Asec(Bx-C)+D
Shifted, Compressed, And/Or Stretched Tangent Function
y=Atan(Bx-C)+D
