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Complete Solutions.
Math 101 Final Review 102 Correct
Complete Solutions.
Definition of a function - ANSWER each input has exactly one output
How to tell if a relation is a function - ANSWER -Passes VLT
-x values do not repeat in ordered pairs
domain - ANSWER x values. Written in interval notation
range - ANSWER y values, written in interval notation
Definition of y-intercept - ANSWER where a graph crosses the y-axis. Where x = 0. (0, y)
How to find the y-intercept - ANSWER Let x = 0 and solve for y. (0, y)
Definition of x-intercept - ANSWER where the graph crosses the x-axis. Where y = 0. (x, 0)
how to find the x intercept(s) of a rational function - ANSWER factor the top, set the numerator = 0 and solve.
increasing interval - ANSWER The interval of x values where the y-values are increasing over the interval. ALWAYS USE PARENTHESIS
decreasing interval - ANSWER The interval of x values where the y-values are decreasing over the interval. ALWAYS USE PARENTHESIS
Complete Solutions.
f(x)=x^2, Quadratic Function (shape & parent points) - ANSWER Parabola, Parent points: (-2, 4)(-1, 1)(0,0)(1, 1)(2, 4)
f(x)= x^3 (cube function) (shape & parent points) - ANSWER Parent points: (-8, -3)(-1, 1)(0,0)(1, 1)(8, 3)
square root function (equation, shape, and parent points) - ANSWER f(x)=√x, Parent points:(0,0)(1, 1)(4, 2)(9, 3)
absolute value function (equation, shape, and parent points) - ANSWER f(x)=|x|, Parent points: (-2, 2)(- 1,1)(0,0)(1,1)(2,1)
exponential function (equation, shape, and parent points) - ANSWER f(x) = b^x, where b>0, and b ≠ 1, has HORIZONTAL asymptotes, Parent points: (0, 1),(1, b)(-1, 1/b)
logarithmic function (equation, shape, and parent points) - ANSWER f(x)=logbX, where b≠1 and b>0, which is the inverse of the exponential function f(x)=b^X. Has VERTICAL asymptotes, Parent points: (1, 0),(b, 1)(1/b, -1)
(f+g)(x)= - ANSWER f(x)+g(x)
(f-g)(x)= - ANSWER f(x)-g(x)
(fg)(x)= - ANSWER f(x) times g(x)
(f/g)(x)= - ANSWER f(x)/g(x)
(f o g)(x)= - ANSWER f(g(x))
In, y=af(b(x-h))+ k, what does A do to the graph? - ANSWER vertical stretch or compression. To apply this to your graph, multiply your Y values by A.
Complete Solutions.
slope-intercept form - ANSWER y=mx+b
m=slope
b=y-intercept
point- slope form of a linear equation - ANSWER (x1, y1) is a point on the line. M is the slope
vertical line equations - ANSWER x = a
horizontal line equation - ANSWER y=b
ZERO SLOPE
Revenue Function - ANSWER R(x) = (# items sold)(price per item)
Cost Function - ANSWER C(x)= Variable Cost + Fixed Cost
break-even point - ANSWER R(x)=C(x)
Revenue = Cost
General form of a quadratic function - ANSWER
vertex form of a quadratic function - ANSWER
axis of symmetry - ANSWER A line that divides the graph into two congruent reflected halves. x = -b/2a or x = h
How to find the vertex of a quadratic (in general form) - ANSWER (-b/2a, f(-b/2a))
Complete Solutions.
How to find the vertex (in vertex form) - ANSWER (h,k)
To find the Maximum/minimum in application problems - ANSWER find the vertex! Maximum/Minimum value is your Y value
completing the square - ANSWER -A process used to form a perfect square trinomial.
-Used to convert from general form to vertex form
-Used to convert an equation into the equation of a circle
Discriminant - ANSWER b²-4ac
how to find the degree of a polynomial in factored form y= (x- )(x- )... - ANSWER add up the exponents on each factor
Turning points in the graph - ANSWER A polynomial of degree n can have at most N-1 turning points
End behavior when LC is positive, degree is even - ANSWER Rises on left and right. Both ends point up
end behavior - ANSWER The behavior of the graph as x approaches positive infinity or negative infinity.
End behavior when LC is negative, degree is even - ANSWER Falls on left and right, both ends point down
End behavior when LC is negative, degree is odd - ANSWER rises on left, falls on right
End behavior when LC is positive, degree is odd - ANSWER falls on left, rises on right
ODD multiplicity of zeros - ANSWER graph crosses x-axis
Complete Solutions.
When is there a slant asymptote? - ANSWER If the degree in the numerator is GREATER than the degree in the denominator
vertical asymptoptes for rational expressions - ANSWER set denominator equal to zero and factor; remember to divide out any common factors from numerator!
Sign test steps - ANSWER 1) find the critical points (vertical asymptotes, x-intercepts)
Build a number line with the critical points to determine possible intervals
Evaluate the function in each interval
Choose the interval(s) for which the inequality is true
Use [ ] if your inequality is EQUAL to
Use ( ) if your inequality is just greater than or less than
Always use ( ) at ASYMPTOTES
direct variation - ANSWER y=kx, k is the constant of variation
Inverse Variation - ANSWER y = k/x, k is the constant of variation
Inverse functions (symmetry) - ANSWER inverse functions are symmetric across the line y=x
Inverse functions (composition) - ANSWER f(f^-1(x)) = f^-1(f(x))
Inverse functions (Domain/Range) - ANSWER Domain of F(x) ----> Range of f^-1(x)
Range of f(x) -----> Domain of f^-1 (x)
Complete Solutions.
Inverse functions (point relationship) - ANSWER if (a, b) is on the graph of f(x) then (b, a) is on the graph of the inverse
To find the inverse function - ANSWER 1. Replace f(x) with y
- Interchange x & y
- Solve the equation for y
- Replace y with f^-1(x)
exponential function (parent graph points) - ANSWER (0, 1) (1, b) (-1, 1/b)
logarithmic function (parent graph points) - ANSWER (1, 0) (b, 1) (1/b, -1)
Compound Interest Formula - ANSWER A= final amount
P=initial amount
r = interest rate
n = # of times compounded per year
t = time
continuous compound interest formula - ANSWER A= final amount
r= rate
t= time
Typical compounding periods - ANSWER Yearly (annually): n= 1
semi-annually n = 2
quarterly n= 4
monthly n = 12
Solutions to a System of Equations (3 cases) - ANSWER
Complete Solutions.
even function symmetry - ANSWER f(x) = f(-x)
y-axis symmetry
(a, b) corresponds to (-a, b)
odd function symmetry - ANSWER f(-x) = -f(x)
Origin of symmetry -- rotate 180 degrees around the origin and its the same.
(a, b) corresponds to (-a, -b)
slope formula - ANSWER
Maximum or minimums of a parabola occur at: - ANSWER the vertex
Multiplicity of a Zero - ANSWER The highest power of (x - r) that appears as a factor of a polynomial.
Complex Zeros Theorem - ANSWER Complex zeros occur in conjugate pairs
ex) if 2 + i is a zero then 2 - i is also a zero
Inverse functions (one- to one relationship) - ANSWER if f(x) has an inverse, both functions must be 1- (pass HLT)
e - ANSWER rational number which is approximately equal to 2.
log with base e - ANSWER ln (x)
Convert exponential to and from logarithmic - ANSWER
