Precalculus Vocab Words
*HOW TO Given the value of a function at different points, calculate the average rate of change of a function for the interval between two values x1 and x2. Section 1.3
1. Calculate the difference y2 - y1 = Δy 2. Calculate the difference x2 - x1 = Δx 3. Find the ratio Δy/Δx
*HOW TO Given a linear function, graph by plotting points. Section 2.2
1. Choose a minimum of two input values. 2. Evaluate the function at each input value. 3. Use the resulting output values to identify coordinate pairs. 4. Plot the coordinate pairs on a grid. 5. Draw a line through the points.
*HOW TO Given a linear function ƒ, and the initial value and rate of change, evaluate ƒ(c).
1. Determine the initial value and the rate of change (slope). 2. Substitute the values into ƒ(x) = mx + b. 3. Evaluate the function at x = c.
*HOW TO Given a complex number, represent its components on the complex plane. Section 3.1
1. Determine the real part and the imaginary part of the complex number. 2. Move along the horizontal axis to show the real part of the number. 3. Move parallel to the vertical axis to show the imaginary part of the number. 4. Plot the point.
*HOW TO Given two points on a line and a third point, write the equation of the perpendicular line that passes through the point. Section 2.2
1. Determine the slope of the line passing through the points. 2. Find the negative reciprocal of the slope. 3. Use the slope-intercept form or point-slope form to write the equation by substituting the known values. 4. Simplify.
*HOW TO Given two points from a linear function, calculate and interpret the slope. Section 2.1
1. Determine the units for output and input values. 2. Calculate the change of output values and change of input values. 3. Interpret the slope as the change in output values per unit of the input value.
*HOW TO Given a tabular function and assuming that the transformation is a vertical stretch or compression, create a table for a vertical compression. Section 1.5
1. Determine the value of a. 2. Multiply all of the output values by a.
*HOW TO Given a polynomial function, determine the intercepts. Section 3.3
1. Determine the y-intercept by setting x = 0 and finding the corresponding output value. 2. Determine the x-intercepts by solving for the input value that yield an output value of zero.
*HOW TO Given two functions ƒ(x) and g(x), test whether the functions are inverses of each other. Section 1.7
1. Determine whether ƒ(g(x)) = x or g(ƒ(x)) = x 2. If both statements are true, then g = ƒ^-1 and ƒ = g^-1. If either statement is false, then both are false, and g ≠ ƒ^-1 and ƒ ≠ g^-1.
*HOW TO Given the formula for a function, determine if the function is even, odd, or neither. Section 1.5
1. Determine whether the function satisfies ƒ(-x) = ƒ(x). If it does, it is even. 2. Determine whether the function satisfies ƒ(-x) = -ƒ(x). If it does, it is odd. 3. If the function does not satisft either rule, it is neither even nor odd.
*HOW TO Given a power function ƒ(x) = kx^n, where n is a non-negative integer, identify the end behavior. Section 3.3
1. Determine whether the power is even or odd. 2. Determine whether the constant is positive or negative. 3. For even function, the positive constant will have both ends heading towards positive infinity. The negative constant will have both ends heading towards negative infinity. 4. For odd function, the positive constant will have the left end heading towards negative infinity, and the right end heading towards positive infinity. The negative constant will have the left end headings towards positive infinity, and the right end heading towards negative infinity.
*HOW TO Given data of input and corresponding outputs from a linear function, find the best fit line using linear regression. Section 2.4
1. Enter the input in List 1 (L1). 2. Enter the output in List 2 (L2). 3. On a graphing utility, select Linear Regression (LinReg).
*HOW TO Given the equation for a linear function, graph the function using the y-intercept and slope.
1. Evaluate the function at an input value of zero to find the y-intercept. 2. Identify the slope as the rate of change of the input value. 3. Plot the point represented by the y-intercept. 4. Use rise/run to determine at least two more points on the line. 5. Sketch the line that passes through the points.
*HOW TO Given a formula for a composite function, evaluate the function. Section 1.4
1. Evaluate the inside function using the input value or variable provided. 2. Use the resulting output as the input to the outside function.
*HOW TO Given a quadratic function ƒ(x), find the y- and x- intercepts. Section 3.2
1. Evaluate ƒ(0) to find the y-intercept. 2. Solve the quadratic equation ƒ(x) = 0 to find the x-intercepts.
*HOW TO Given the formula for a function, determine the domain and range. Section 1.2
1. Exclude from the domain any input values that result in division by zero. 2. Exclude from the domain any input values that have nonreal (or undefined) number outputs. 3. Use the valid input values to determine the range of the output values. 4, Look at the function graph and table values to confirm the actual function behavior.
*HOW TO Given an absolute value inequality of the form |x - A| ≤ B for real numbers a and b where b is positive, solve the absolute value inequality algebraically. Section 1.6
1. Find boundary points by solving |x - A| = B. 2. Test intervals created by the boundary points to determine where |x - A| ≤ B. 3. Write the interval or union of intervals satisfying the inequality in interval, inequality, or set-builder notation.
*HOW TO Given the graph of a function, evaluate its inverse at specific points. Section 1.7
1. Find the desired input on the y-axis of the given graph. 2. Read the inverse function's output from the x-axis of the given graph.
*HOW TO Given a function composition ƒ(g(x)), determine its domain. Section 1.4
1. Find the domain of g. 2. Find the domain of ƒ. 3. Find those inputs x in the domain of g for which g(x) is in the domain of ƒ. That is, exclude those inputs x from the domain of g for which g(x) is not in the domain of ƒ. The resulting set is the domain of ƒ∘g.
*HOW TO Given a function represented by a table, identify specific output and input values. Section 1.1
1. Find the given input in the row (or column) of input values. 2. Identify the corresponding output value paired with that input value. 3. Find the given output values in the row (or column) of output values, noting every time that output value appears. 4. Identify the input value(s) corresponding to the given output value.
*HOW TO Given a polynomial function, identify the degree and leading coefficient. Section 3.3
1. Find the highest power of x to determine the degree of the function. 2. Identify the term containing the highest power of x to find the leading term. 3. Identify the coefficient of the leading term.
*HOW TO Given a polynomial function, sketch the graph. Section 3.4
1. Find the intercepts. 2. Check for symmetry. If the function is an even function, its graph is symmetrical about the y-axis, that is, ƒ(-x) = ƒ(x). If a function is an odd function, its graph is symmetrical about the origin, that is, ƒ(-x) = -ƒ(x). 3. Use the multiplicities of the zeros to determine the behavior of the polynomial at the x-intercepts. 4. Determine the end behavior by examining the leading term. 5. Use the end behavior and the behavior at the intercepts to sketch a graph. 6. Ensure that the number of turning points does not exceed one less than the degree of the polynomial. 7. Optionally, use technology to check the graph.
*HOW TO Given the equation of a function and a point through which its graph passes, write the equation of a line perpendicular to the given line. Section 2.2
1. Find the slope of the function. 2. Determine the negative reciprocal of the slope. 3. Substitute the new slope and the values for x and y from the coordinate pair provided into g(x) = mx + b. 4. Solve for b. 5. Write the equation for the line.
*HOW TO Given the equation of a function and a point through which its graph passes, write the equation of a line parallel to the given line that passes through the given point. Section 2.2
1. Find the slope of the function. 2. Substitute the given values into either the general point-slope equation of the slope-intercept equation for a line. 3. Simplify.
*HOW TO Given the equation of a linear function , use transformations to graph the linear function in the form ƒ(x) = mx + b Section 2.2
1. Graph ƒ(x) = x. 2. Vertically stretch or compress the graph by a factor m. 3. Shift the graph up or down b units.
*HOW TO Given a quadratic function in general form, find the vertex of the parabola. Section 3.2
1. Identify a, b, & c. 2. Find h, the x-coordinate of the vertex, by substituting a & b into h = -b/2a. 3. Find k, the y-coordinate of the vertex, by evaluating k = ƒ(h) = ƒ(-b/2a)
*HOW TO Given a function written in equation form including an even root, find the domain. Section 1.2
1. Identify input values. 2. Since there is an even root, exclude any real numbers that result in a negative number in the radicand. Set the radicand greater than or equal to zero and solve for x. 3. The solution(s) are the domain of the function. If possible, write the answer in interval form.
*HOW TO Given a quadratic function, find the domain and range. Section 3.2
1. Identify the domain of any quadratic function as all real numbers. 2. Determine whether a is positive or negative. If a is positive, the parabola has a minimum. If a is negative, the parabola has a maximum. 3. Determine the maximum or minimum value of the parabola, k. 4. If the parabola has a minimum, the range is given by ƒ(x) ≥ k OR [k, ∞) If the parabola has a maximum, the range is given by ƒ(x) ≤ k OR (-∞, k]
*HOW TO Given a graph of a quadratic function, write the equation of the function in general form. Section 3.2
1. Identify the horizontal shift of the parabola; this value is h. Identify the vertical shift of the parabola; this value is k. 2. Substitute the values of the horizontal and vertical shift for h and k in the function ƒ(x) = a(x-h)^2 + k. 3. Substitute the values of any point, other than the vertex, on the graph of the parabola for x and ƒ(x). 4. Solve for the stretch factor, |a|. 5. If the parabola opens up, a > 0. If the parabola opens down, a < 0 since this means the graph was reflected about the x-axis. 6. Expand and simplify to write in general form.
*HOW TO Given a situation that represents a system of linear equations, write the system of equations and identify the solution. Section 2.3
1. Identify the input and output of each linear model. 2. Identify the slope and y-intercept of each linear model. 3. Find the solution by setting the two linear functions equal to one another and solving for x, or find the point of intersection on a graph.
*HOW TO Given a table of input and output values, determine whether the table represents a function. Section 1.1
1. Identify the input and output values. 2. Check to see if each input value is paired with only one output value. If so, the table represents a function.
*HOW TO Given a word problem that includes two pairs of input and output values, use the linear function to solve a problem. Section 2.3
1. Identify the input and output values. 2. Convert the date to two coordinate pairs. 3. Find the slope. 4. Write the linear model. 5. Use the model to make a prediction by evaluating the function at a given x-value. 6. Use the model to identify an x-value that results in a given y-value. 7. Answer the question posed.
*HOW TO Given a tabular function, creat a new row to represent a horizontal shift. Section 1.5
1. Identify the input row or column. 2. Determine the magnitude of the shift. 3. Add the shift to the value in each input cell.
*HOW TO Given a function written in equation form, find the domain. Section 1.2
1. Identify the input values. 2. Identify any restrictions of the input and exclude those values from the domain. *(neg. # under a radical with an even index; division by 0) 3. Write the domain in interval form, if possible.
*HOW TO Given a function written in an equation form that includes a fraction, find the domain. Section 1.2
1. Identify the input values. 2. Identify any restrictions on the input. If there is a denominator in the function's formula, set the denominator equal to zero and solve for x. If the function's formula contains an even root, set the radicand greater than of equal to 0, and then solve. 3. Write the domain in interval form, making sure to exclude any restricted values from the domain.
*HOW TO Given a relationship between two quantities, determine whether the relationship is a function. Section 1.1
1. Identify the input values. 2. Identify the output values. 3. If each input value leads to only one output value, classify the relationship as a function. If any input value leads to two or more outputs, do not classify the relationship as a function.
*HOW TO Given a piecewise function, write the formula and identify the domain for each interval. Section 1.2
1. Identify the intervals for which different rules apply. 2. Determine formulas that describe how to calculate an output from an input in each interval. 3. Use braces and if-statements to write the function.
*HOW TO Given a line graph, describe the set of values using interval notation. Section 1.2
1. Identify the intervals to be included in the set by determining where the heavy line overlays the real line. 2. At the left end of each interval, use [ with each end value to be included in the set (solid dot) or ( for each excluded end value (open dot). 3. At the right end of each interval, use ] with each end value to be included in the set (filled dot) or ) for each excluded end value (open dot). 4. Use the union symbol ∪ to combine all intervals into one set.
*HOW TO Given a tabular function, create a new row to represent a vertical shift. Section 1.5
1. Identify the output row or column. 2. Determine the magnitude of the shift. 3. Add the shift to the value in each output cell. Add a positive value for up or a negative value for down.
*HOW TO Given two complex numbers, find the sum or difference. Section 3.1
1. Identify the real and imaginary parts of each number. 2. Add or subtract the real parts. 3. Add or subtract the imaginary parts.
*HOW TO Given a function, graph its vertical stretch. Section 1.5
1. Identify the value of a. 2. Multiply all range values by a. 3. • If a > 1, the graph is stretched by a factor of a. • If 0 < a < 1, the graph is compressed by a factor of a. • If a < 0, the graph is either stretched or compressed and also reflected about the x-axis.
*HOW TO Given a function and both a vertical and a horizontal shift, sketch the graph. Section 1.5
1. Identify the vertical and horizontal shifts from the formula. 2. The vertical shift results from a constant added to the output. Move the graph up for a positive constant and down for a negative constant. 3. The horizontal shift results from a constant added to the input. Move the graph left for a positive constant and right for a negative constant. 4. Apply the shifts to the graph in either order.
*HOW TO Given a graph of linear function, find the equation to describe the function. Section 2.2
1. Identify the y-intercept of an equation. 2. Choose two points to determine the slope. 3. Substitute the y-intercept and slope into the slope-intercept form of a line.
*HOW TO Given the graph of a linear function, write an equation to represent the function. Section 2.1
1. Identify two points on the line. 2. Use the two points to calculate the slope. 3. Determine where the line crosses the y-axis to identify the y-intercept by visual inspection. 4. Substitute the slope and y-intercept into the slope-intercept form of a line equation.
*HOW TO Given a function, find the domain and range of its inverse. Section 1.7
1. If the function is one-to-one, write the range of the original function as the domain of the inverse, and write the domain of the original function as the range of the inverse. 2. If the domain of the original function needs to be restricted to make it one-to-one, then this restricted domain becomes the range of the inverse function.
*HOW TO Given a graph of a polynomial function of degree n, identify the zeros and their multiplicities
1. If the graph crosses the x-axis and appears almost linear at the intercept, it is a single zero. 2. If the graph touches the x-axis and bounces off of the axis, it is a zero with an even multiplicity. 3. If the graph crosses the x-axis at a zero, it is a zero with off multiplicity. 4. The sum of the multiplicities is n.
*HOW TO Given a piecewise function, sketch a graph. Section 1.2
1. Indicate on the x-axis the boundaries defined by the intervals on each piece of the domain. 2. For each piece of the domain, graph on that interval using the corresponding equation pertaining to that piece. Do not graph two functions over one interval because it would violate the criteria of a function.
*HOW TO Given a graph of a function, use the horizontal line test to determine if the graph represents a one-to-one function. Section 1.1
1. Inspect the graph to see if any horizontal line drawn would intersect the curve more than once. 2. If there is any such line, determine that the function is not one-to-one.
*HOW TO Given a graph, use the vertical line test to determine if the graph represents a function. Section 1.1
1. Inspect the graph to see if any vertical line drawn would intersect the curve more than once. 2. If there is any such line, determine that the graph does not represent a function.
*HOW TO Given an absolute value equation, solve it. Section 1.6
1. Isolate the absolute value term. 2. Use |A| = B to write A = B or -A = B, assuming B > 0. 3. Solve for x.
*HOW TO Given the formula for an absolute value function, find the horizontal intercepts of its graph. Section 1.6
1. Isolate the absolute value term. 2. Use |A| = B to write A = B or -A = B, assuming B > 0. 3. Solve for x.
*HOW TO Given a composite function and graphs of its individual functions, evaluate it using the information provided by the graphs. Section 1.4
1. Locate the given input to the inner function on the x-axis of its graph. 2. Read off the output of the inner function from the y-axis of its graph. 3. Locate the inner function output on the x-axis of the graph of the outer function. 4. Read the output of the outer function from the y-axis of its graph. This is the output of the composite function.
*HOW TO Given a function represented by a formula, find the inverse. Section 1.7
1. Make sure ƒ is a one-to-one function. 2. Solve for x. 3. Interchange x and y.
*HOW TO Given a function, reflect the graph both vertically and horizontally. Section 1.5
1. Multiply all the outputs by -1 for a vertical reflection. The new graph is a reflection of the original graph about the x-axis. 2. Multiply all inputs by -1 for a horizontal reflection. The new graph is a reflection of the original graph about the y-axis.
*HOW TO Given an absolute value function, solve the set of inputs where the output is positive (or negative). Section 1.6
1. Set the function equal to zero, and solve for the boundary points of the solution set. 2. Use test points or a graph to determine where the function's output is positive or negative.
*HOW TO Given a polynomial function ƒ, find the x-intercepts by factoring. Section 3.4
1. Set ƒ(x) = 0 2. If the polynomial function is not given in factored form: a. Factor out any common monomial factors b. Factor any factorable binomials or trinomials 3. Set each factor equal to zero and solve to find the x-intercepts.
*HOW TO Given a function in equation form, write its algebraic formula. Section 1.1
1. Solve the equation to isolate the output variable on one side of the equal sign, with the other side as an expression that involves only the input variable. 2. Use all the usual algebraic methods for solving equations, such as adding or subtracting the same quantity to or from both sides, or multiplying or dividing both sides of the equation by the same quantity.
*HOW TO Given a quadratic function, find the x-intercepts by rewriting in standard form Section 3.2
1. Substitute a and b into h = -b/2a 2. Substitute x = h into the general form of the quadratic function to find k. 3. Rewrite the quadratic in standard form using h and k. 4. Solve for when the output of the function will be zero to find the x-intercepts.
*HOW TO Given the formula for a function, evaluate. Section 1.1
1. Substitute the input variable in the formula with the value provided. 2. Calculate the result *(plug & chug)
Parallel and Perpendicular Lines Section 2.2
1. Two lines are parallel lines if they do not intersect and the slopes of the lines are the same. m1 = m2 2. Two lines are perpendicular lines if they intersect at right angles, and the slopes are opposite reciprocals. m1 • m2 = -1
*HOW TO Given two complex numbers, multiply to find the product. Section 3.1
1. Use the distributive property or the FOIL method. 2. Simplify.
*HOW TO Given a complex number and a real number, multiply to find the product. Section 3.1
1. Use the distributive property. 2. Simplify.
*HOW TO Given an imaginary number, express it in standard form. Section 3.1
1. Write (√-a) as (√a)(√-1). 2. Express (√-1) as i. 3. Write (√a) • i in simplest form.
*HOW TO Given a description of a function, sketch a horizontal compression or stretch. Section 1.5
1. Write a formula to represent the function. 2. Set g(x) = ƒ(bx) where b > 1 for a compression or 0 < b < 1 for a stretch.
*HOW TO Given an application involving revenue, use a quadratic equation to find the maximum. Section 3.2
1. Write a quadratic equation for revenue. 2. Find the vertex of the quadratic equation. 3. Determine the y-value of the vertex.
*HOW TO Given two complex numbers, divider one by the other. Section 3.1
1. Write the division problem as a fraction. 2. Determine the complex conjugate of the denominator. 3. Multiply the numerator and denominator of the fraction by the complex conjugate of the denominator. 4. Simplify.
Imaginary & Complex Numbers Section 3.1
A complex number is a number of the form a + bi where • a is the real part of the complex number. • bi is the imaginary part of the complex number. If b = 0, then a + bi is a real number. If a = 0 and b is not equal to 0, the complex number is called an imaginary number. An imaginary number is an even root of a negative number.
Decreasing Function Section 1.3
A function f is a decreasing function on an open interval if ƒ(b) < ƒ(a) for every a, b interval where b > a.
Function Section 1.1
A function is a relation in which each possible input value leads to exactly one output value. We say "the output is a function of the input." The input values make up the domain. The output values make up the range. *(vertical line test)
Even Function Section 1.5
A function is called an even function if for every input x ƒ(-x) = ƒ(x) The graph of an even function is symmetric about the y-axis.
Odd Function Section 1.5
A function is called an odd if for every input x ƒ(-x) = -ƒ(x) The graph of an odd function is symmetric about the origin.
Local Minima & Local Maxima Section 1.3
A function ƒ has a local minimum at a point b in an open interval (a, c) if ƒ(b) is less than or equal to ƒ(x) for every point x (x does not equal b) in the interval. A function ƒ has a local maximum at a point b in an open interval (a, c) if ƒ(b) is greater than or equal to ƒ(x) for every point x (x does not equal b) in the interval.
Increasing Function Section 1.3
A function ƒ is an increasing function on an open interval if ƒ(b) > ƒ(a) for every a, b interval where b > a.
Linear Function Section 2.1
A linear function is a function whose graph is a line. Linear functions can be written in the slope-intercept form of a line ƒ(x) = mx + b Where b is the initial or starting value of the function (when input, x = 0), and m is the constant rate of change, or slope of the function. The y-intercept is at (0,b)
One-To-One Function Section 1.1
A one-to-one function is a function in which each output value corresponds to exactly one input value. *(horizontal line test)
Piecewise Function Section 1.2
A piecewise function is a function in which more than one formula is used to define the output. Each formula has its own domain, and the domain of the function is the union of all these smaller domains. ƒ(x) = {formula 1 if x is in domain 1 {formula 2 if x is in domain 2 {formula 3 if x is in domain 3
Intercepts & Turning Points of Polynomials Section 3.3
A polynomial of degree n will have, at most, n x-intercepts and n - 1 turning points.
Power Function Section 3.3
A power function is a function that can be represented in the form ƒ(x) = kx^p where k and p are real numbers, and k is known as the coefficient.
Forms of Quadratic Functions Section 3.2
A quadratic function is a function of degree two. The graph of a quadratic function is a parabola. The general form of a quadratic function is ƒ(x) = ax^2 + bx + c where a, b, and c are real numbers and a ≠ 0. The standard form of a quadratic function is ƒ(x) = a(x-h)^2 + k The vertex (h,k) is located at h = -b/2a k = ƒ(h) = ƒ(-b/2a)
Rate of Change Section 1.3
A rate of change describes how an output quantity changes relative to the change in the input quantity. The units on a rate of change are "output units per input units." The average rate of change between two input values is the total change of the function values (output values) divided by the change in input values. *(slope, rise over run)
Interpreting Turning Points Section 3.4
A turning point is a point of the graph where the graph changes from increasing to decreasing (rising to falling) or decreasing to increasing (falling to rising). A polynomial of degree n will have at most n - 1 turning points.
Intercepts & Turning Points of Polynomial Functions Section 3.3
A turning point of a graph is a point at which the graph changes direction from increasing to decreasing or decreasing to increasing. The y-intercept is the point at which the function has an input value of zero. The x-intercepts are the points at which the output value is zero.
Complex Numbers: Addition & Subtraction Section 3.1
Adding complex numbers: (a + bi) + (c + di) = (a + c) + (b + d) • i Subtracting complex numbers: (a + bi) - (c + di) = (a - c) + (b - d) • i
Interpolation & Extrapolation Section 2.4
Different methods of making predictions are used to analyze data. • The method of interpolation involves predicting a value inside the domain and/or range of the data. • The method of extrapolation involves predicting a value outside the domain and/or range of the data. • Model Breakdown occurs at the point when the model no longer applies.
Inverse Function Section 1.7
For any one-to-one function ƒ(x) = y, a function ƒ^-1(x) is an inverse function of ƒ if ƒ^-1(y) = x. This can also be written as ƒ^-1(ƒ(x)) = x for all x in the domain of ƒ. It also follows that ƒ(ƒ^-1(x)) = x for all x in the domain of ƒ^-1 if ƒ^-1 is the inverse of ƒ. The notation ƒ^-1 is read "ƒ inverse." Like any other function, we can use any variable name as the input for ƒ^-1, so we will often write ƒ^-1(x), which we read as "ƒ inverse of x." Inverse functions have symmetry about the line y=x. Keep in mind that ƒ^-1(x) ≠ 1/ƒ(x) & that not all functions have inverses.
Solutions to Absolute Value Equations Section 1.6
For real numbers A & B, with an equation of the form |A| = B If B ≥ 0, A = B or A = -B If B < 0, the equation |A| = B has no solution.
Vertical Stretches & Compressions Section 1.5
Given a function ƒ(x), a new function g(x) = aƒ(x) where a is a constant, is a vertical stretch or vertical compression of the function ƒ(x). • If a > 1, then the graph will be stretched. • If 0 < a < 1, then the graph will be compressed. • If a < 0, then there will be combination of a vertical stretch or compression with a vertical reflection.
Horizontal Reflection Section 1.5
Given a function ƒ(x), a new function g(x) = ƒ(-x) is a horizontal reflection of the function ƒ(x), sometimes called a reflection about the y-axis.
Horizontal Stretches & Compressions Section 1.5
Given a function ƒ(x), a new function g(x) = ƒ(bx) where b is a constant, is a horizontal stretch or horizontal compression of the function ƒ(x). • If b > 1, then the graph will be compressed by 1/b. • If 0 < b < 1, then the graph will be stretched by 1/b. • If b < 0, then there will be a combination of a horizontal stretch or compression with a horizontal reflection
Vertical Reflection Section 1.5
Given a function ƒ(x), a new function g(x) = -ƒ(x) is a vertical reflection of the function ƒ(x), or sometimes called reflection about the x-axis.term-13
Vertical Shift Section 1.5
Given a function ƒ(x), a new function g(x) = ƒ(x) + d where k is a constant, is a vertical shift of the function ƒ(x). All the output values change by k units. If d is positive, the graph will shift up. If d is negative, the graph will shift down.
Horizontal Shift Section 1.5
Given a function ƒ, a new function g(x) = ƒ(x - c) where c is a constant, is a horizontal shift of the function ƒ. If c is positive, the graph will shift right. If c is negative, the graph will shift left. *[If g(x) = ƒ(bx - c), there will be a horizontal shift of c/b units]
Graphical Behavior of Polynomials at x-Intercepts Section 3.4
If a polynomial contains a factor of the form (x - h)^p, the behavior near the x-intercept h is determined by the power p. We say that x = h is a zero of multiplicity p. The graph of a polynomial function will touch the x-axis at zeros with even multiplicities. The graph will cross the x-axis at zeros with off multiplicities. The sum of the multiplicities is the degree of the polynomial function.
Complex Plane Section 3.1
In the complex plane, the horizontal axis is the real axis, and the vertical axis is the imaginary axis.
Graphical Interpretation of a Linear Function Section 2.2
In the equation ƒ(x) = mx + b • b is the y intercept of the graph and indicates the point (0,b) at which the graph crosses the y-axis. • m is the slope of the line and indicates the vertical displacement (rise) and horizontal displacement (run) between each successive pair of points. Recall the formula for slope: change in y over change in x.
Interval Notation Section 1.2
Interval notation is a way of describing sets that include all real numbers between a lower limit that may or may not be included and an upper limit that may or may not be included. The endpoint values are listed between brackets or parentheses. A square bracket indicates inclusion in the set, and a parenthesis indicates exclusion from the set. Ex. (4, 12]
Polynomial Functions Section 3.3
Let n be a non-negative integer. A polynomial function is a function that can be written in the form ƒ(x) = a_n•x^n + a_(n-1)•x^(n-1) + ... + a_2•x^2 + a_1•x + a_0 This is called the general form of a polynomial function. Each a_i is a coefficient and can be any real number, but a_n ≠ 0. Each product a_i•x^1 is a term of a polynomial function.
Horizontal and Vertical Lines Section 2.2
Lines can be horizontal or vertical. 1. A horizontal line is a line defined by an equation in the form ƒ(x) = b 2. A vertical line is a line defined by an equation in the form x = a
Set-Builder Notation Section 1.2
Set-builder notation is a method of specifying a set of elements that satisfy a certain condition. It takes the form {x| statement about x} which is read as, "the set of all x such that the statement about x is true." Ex. {x| 4 < x ≤ 12}
Absolute Minima & Maxima Section 1.3
The absolute minimum of ƒ at x=c is ƒ(c) where ƒ(c) ≤ ƒ(x) for all x in the domain of ƒ. The absolute maximum of ƒ at x=d is ƒ(d) where ƒ(d) ≥ ƒ(x) for all x in the domain of ƒ.
Absolute Value Function Section 1.6
The absolute value function can be defined as a piecewise function f(x)=|x|= {x if x ≥ 0 {−x if x < 0
The Complex Conjugate Section 3.1
The complex conjugate of a complex number is a + bi & a - bi It is found by changing the sign of the imaginary part of the complex number. The real part of the number is left unchanged. • When a complex number is multiplied by its complex conjugate, the result is a real number. • When a complex number is added to its complex conjugate, the result is a real number.
Correlation Coefficient Section 2.4
The correlation coefficient is a value, r, between -1 & 1. • r > 0 suggests a positive (increasing) relationship • r < 0 suggests a negative (decreasing) relationship • The closer the value is to 0, the more scattered the data. • The closer the value is to 1 or -1, the less scattered the data.
Domain of a Composite Function Section 1.4
The domain of a composite function ƒ(g(x) is the set of those inputs x in the domain of g for which g(x) is in the domain of ƒ.
Function Notation Section 1.1
The notation y=ƒ(x) defines a function named ƒ. This is read as "y is a function of x." The letter x represents the input value, or independent variable. The letter y, or ƒ(x), represents the output value, or dependent variable.
Point- Slope Form of a Linear Equation Section 2.1
The point-slope form of a linear equation takes the form y − y1 = m(x − x1) where m is the slope, x1 and y1 are the x- and y- coordinates of a specific point through which the line passes.
Domain & Range of Inverse Functions Section 1.7
The range of a function ƒ(x) is the domain of the inverse function f^-1(x). The domain of ƒ(x) is the range of f^-1(x).
Increasing & Decreasing Functions 2.1
The slope determines if the function is an increasing linear function, a decreasing linear function, or a constant function. •ƒ(x) = mx + b increasing if m > 0 •ƒ(x) = mx + b decreasing if m < 0 •ƒ(x) = mx + b constant if m = 0
X-Intercept Section 2.2
The x-intercept of the function is value of x when ƒ(x) = 0. It can be solved by the equation 0 = mx + b
Terminology of Polynomial Functions Section 3.3
We often rearrange polynomials so that the powers are descending. When a polynomial is written in this way, we say that it is in general form. The a_i•x^i with the highest exponent is called the 'leading term'. The constant a_i is called the 'leading coefficient'. the exponent i on x^i is the 'degree' of the polynomial.
Composition of Functions Section 1.4
When the output of one function is used as the input of another, we call the entire operation a composition of functions. For any input x and functions ƒ and g, this action defines a composite function. Which we write as ƒ∘g such that (ƒ∘g)(x) = ƒ(g(x)) The domain of the composite function ƒ∘g is all x such that x is in the domain of g and g(x) is in the domain of ƒ. It is important to realize that the product of functions ƒg is not the same as the function composition ƒ(g(x)), because, in general, ƒ(x)g(x) ≠ ƒ(g(x)).
Domain & Range of a Quadratic Function Section 3.2
• The domain of any quadratic function is all real numbers. • The range of a quadratic function written in general form ƒ(x) = ax^2 + bx + c with a positive a value is ƒ(x) ≥ ƒ(-b/2a) OR [ ƒ(-b/2a), ∞) • The range of a quadratic function written in general form ƒ(x) = ax^2 + bx + c with a negative a value is ƒ(x) ≤ ƒ(-b/2a) OR (-∞, ƒ(-b/2a)] • The range of a quadratic function written in standard form ƒ(x) = a(x-h)^2 + k with a positive a value is ƒ(x) ≥ k; the range of a quadratic function written in standard for mwith a negative a value is ƒ(x) ≤ k.
