Math 1021 Final Review
vertical line test
A graph in the Cartesian plane is the graph of a function if every vertical line intersects the graph no more than once.
solving a polynomial equation of the form *ax^3 +bx^2 +cx*
1) Set the equation equal to zero and factor out an x 2) subtract ax^3 and bx^2 from both sides 3) subtract cx from both sides
3 ways to describe the solutions to a linear inequality
1) Write the solution in set-builder notation. 2) Write the solution in interval notation. 3) Graph the solution on a number line.
Slope of a line
(y2-y1)/(x2-x1) = Rise/Run = change in y over change in x
Which of the following statements most commonly describes the relationship between the quantity x of a product and the price p?
As the quantity x increases, the price p tends to decrease.
Which of the following best describes the zero product property?
Which of the following best describes the zero product property?
Many functions have restricted domains.
true
If u is an algebraic expression and c is a real number such that c>0, then the equation |u = c| is equivalent to
u = −c or u = c.
vertical shift
up = f(x) + c down = f(x) - c
f(x) = ax^n
- If a is positive and n is odd, the graph approaches negative infinity on the left side and positive infinity on the right side. - If a is positive and n is even, the graph approaches positive infinity on the left side and positive infinity on the right side. - If n=1, the graph is a straight line.
the distance between two points
- always greater than or equal to zero - cannot be negative
y = logb(x) for b > 1?
- containts the point (b,1) - contains the point (1,0) - x=0 asymptote
Properties of Inequalities
- for c < 0, if a < b, a/c > b/c - for c < 0, if a < b, ac > bc - for c > 0, if a < b, ac < bc
revenue business model
- ideally, the graph of a simple revenue function should have a maximum value - when the demand equation is linear, it can represented by p=mx+b, where x is the quantity sold and p is the price.
which functions increase on the interval (-∞, ∞)
- identity - cubic - cube root - ??? any more?
Horizonal asymptotes of f(x) = g(x) / h(x)
- if degree of g < h, then y=0 - if degree of g = h, then y= g/h - if degree of g > h, no horizontal asymptote
graph of f(x) = e^x
- intersects the y-axis at (0,1). - approaches 0 as x approaches negative infinity. - y=0 asymptote
If u is an algebraic expression and c is a real number such that c>0, then the inequality |u| < c is equivalent to
-c < u < c
The rectangular coordinate system is also called the
Cartesian coordinate system
Which of the following is typically NOT one of the three ways to describe the solutions to a linear inequality?
Write the solution in inequality form.
removable discontinuity
a "hole" in a graph - g(x) and h(x) share a common factor
relation
a correspondence between two sets A and B such that each element of set A corresponds to one or more elements in set B
function
a relation such that for each element in the domain, there is exactly one corresponding element in the range
Given an ordered pair of the form (x,y), x is called the
abscissa.
quadratic function domain
all real numbers
even function
graph is symmetrical with respect to the y-axis; f(x) = f(-x) *x values change
If two distinct nonvertical lines are parallel, then the two lines must
have the same slope
A rational function may have many vertical asymptotes but may NOT have many
horizontal asymptotes
y = b
horizontal line with zero slope
Given any two distinct lines in the Cartesian plane, the two lines will either
intersect or they will be parallel
number e
irrational number
horizontal shift
left = f(x + c) right = f(x - c)
logarithm power rule
logb(U)^r = r logb(U)
logarithm quotient rule
logb(U/V) = logb(U) - logb(V)
logarithm product rule
logb(UV) = logb(U) + logb(V)
The ratios of the vertical rise to the horizontal run of any two distinct nonvertical parallel lines
must be equal
Discriminant = negative
no real solutions (2 imaginary solutions)
Given an ordered pair of the form (x,y), y is called the
ordinate
inverse function graph
over line y = x
midpoint formula
[(x1+x2)/2, (y1+y2)/2] where the endpoints are (x1, y1) and (x2, y2).
vertex form
f(x)=a(x-h)^2+k
point-slope form
y - y₁ = m (x - x₁)
even parent functions (symmetric about y-axis)
y = 0 (straight line) y = x^2 (quadratic) y = |x| (absolute value)
odd parent functions (symmetric about origin)
y = 1/x (reciprocal) y = x (identity) y = 3√x (cube root) y = x^3 (cubic)
Logarithmic to exponential form
y = logb(x) = x = b^y
It is NOT possible for a piecewise-defined function to have more than one
y-intercept (depending on how the function is defined)
A function canNOT have several
y-intercepts
Discriminant = positive
2 real solutions
profit function
P(x) = R(x) - C(x)
revenue function
R(x) = (# items sold)(price per item)
discriminant = 0
one real solution
Which of the following statements IS true concerning the equation *x^2 - c = 0* for *c > 0*
1) A quadratic equation in this form can always be solved using the square root property. 2) A quadratic equation in this form can always be solved by factoring. 3) The left-hand side of this equation is called a difference of two squares.
Which of the following statements is NOT true about the revenue business model?
A revenue function can never be defined as a function of the price p.
Which of the following statements best defines the term "extraneous solution"?
An extraneous solution is a solution obtained through algebraic manipulations that is not a solution to the original equation.
Which of the following statements about projectile motion is NOT TRUE?
An object thrown or shot vertically into the air reaches a maximum height after t seconds (when time is measured in seconds), where t is the k-coordinate of the vertex of the parabola.
Which of the following is NOT a valid strategy when solving a polynomial equation of the form *ax^3 +bx^2 +cx*?
Divide the equation by x.
The rectangular coordinate system is divided into four quadrants labeled as quadrants
I, II, III, and IV
Which of the following statements is not true? (one-to-one functions, inverse)
If a function f has an inverse function, then we can find the inverse function by replacing f(x) with y, interchanging the variables x and y, and solving for x.
Every one-to-one function has an inverse function.
If f and f^−1 are inverse functions, then the domain of f is the same as the range of f^−1 If f and f^-1 are inverse functions and f(a)=b, then f^-1(b) = a
Which of the following statements is NOT true? (functions)
If the domain of a function consists of more than one element, then the range must also consist of more than one element.
Which of the following statements is true about vertical asymptotes of a rational function of the form f(x) = g(x) / h(x) where g and h are polynomial functions?
In order to correctly determine the vertical asymptotes, it is essential to cancel any common factors of g and h.
Which of the following statements is true about rational equations?
It is important to check the solutions to a rational equation because it is possible to encounter extraneous solutions.
Let line 1 with slope m1 and line 2 with slope m2 be two *nonvertical perpendicular* lines. Which of the following statements is NOT true?
It is possible for the value of m1 to be zero.
If an exponential equation can be written in the form b^u=b^v, then which of the following methods may be used to solve the equation?
Relating the bases by setting u=v.
where is the distance formula derived from?
The Pythagorean theorem
In the definition of the exponential function f(x)=b^x what are the stipulation(s) for the base b?
The base b must be greater than zero and not equal to 1.
Given two points which represent the endpoints of the diameter of a circle, which of the following statements is true?
The center of the circle is the midpoint of the given endpoints of the diameter.
Which of the following statements is true about linear equations of the form *ax + b = 0*?
The constants a and b must be real numbers with the constant a never equal to zero.
Which of the following statements describes the absolute value of a number a?
The distance from the number a to 0 on a number line can be represented by the absolute value of a number a.
Which of the following statements is not true about a rational function of the form f(x) = g(x) / h(x) where g and h are polynomial functions?
The domain of f(x) consists of all values of x such that g(x)≠0 and h(x)≠0.
Which statement describes how to derive the equation of a circle in standard form?
The equation of a circle can be derived using the distance formula.
Which of the following statements is NOT true about the profit business model?
The revenue is always more than the cost.
Given the graph of a quadratic function with the vertex and the y-intercept clearly identified, which of the following statements is NOT TRUE?
The value of b in f(x)=ax2+bx+c can easily be determined from the shape of the graph.
Suppose that you are solving a quadratic equation and realize that the discriminant is equal to 7. Which of the following statements best describes the solutions to this quadratic equation?
There must be two real solutions.
Which of the following statements is NOT true concerning the equation *x^2 - c = 0* for *c > 0*
This equation is not considered to be a quadratic equation because it is not of the form ax^2 + bx + c
Which of the following statements is true about the quadratic function f(x) = ax² + bx + c
constants a, b, and c must be real numbers with a not ever equal to zero. - c is y-intercept of graph
NOT true about graph of y = logb(x) for b > 1?
decreasing on the interval (0,∞).
the sign of the leading coefficient a
determines right hand behavior of graph of polynomial function
one-to-one function
each element of the domain pairs to exactly one unique element of the range (passes horizontal AND vertical line tests)
The shape of the graph of a polynomial function near the x-intercepts can be determined by
examining the multiplicity of the real zeros.
f(x) = |x|
f(x) = {x if x ≥ 0 AND -x if x < 0}
A function is increasing on an interval (a,b) if, for any x1 and x2 chosen from the interval with x1< x2, then
f(x1) < f(x2)
Which of the following is NOT a property of inequalities?
for c < 0, if a < b, a-c > b-c
The domain of f(x) = logb[g(x)]
g(x) > 0
Which of the following statements is not true for the graph of f(x)=b^x where 0 < b < 1 ?
graph approaches 0 as x approaches negative infinity.
odd function
graph is symmetrical with respect to the origin f(-x) = -f(x) *function sign changes
X-Intercepts of a Quadratic
the real zeros of the function
All linear equations and all quadratic equations are polynomial equations.
true
Every function is a relation but not every relation is a function.
true
If n≥2 is an EVEN integer, then the domain of f(x)=n√g(x) is the solution to the inequality g(x) ≥ 0.
true
If the domain and range of a relation are sets of real numbers, then the relation can be represented by plotting ordered pairs in the Cartesian plane.
true
It is possible for an absolute value equation to have no solution.
true
The domain of every polynomial function is all real numbers.
true
To verify that two one-to-one functions, f and g, are inverses of each other, we must show that f(g(x))=g(f(x))=x.
true
Two or more distinct elements in the domain of a function can correspond to the same element in the range.
true
If u is an algebraic expression and c is a real number such that c>0, then the inequality |u| > c is equivalent to
u < −c or u > c.
f(x)= a(x-h)^2 + k
vertex coordinate is (h,k) - a determines shape of graph
logb(b^x)
x
axis of symmetry
x = h h is the x-coordinate of the vertex.
A function can have infinitely many
x-intercepts
