Functions & Transformations
What does it mean for a function to be even?
Answer: f(−x) = f(x)
What does the solution to a system represent graphically?
The point(s) where graphs intersect
Where is the maximum or minimum of a quadratic found?
At the vertex
What determines the end behavior of a polynomial?
The leading term
What causes a vertical asymptote?
Denominator equals zero
Over what interval is y=∣x+2∣ increasing?
(−2,∞)
Solve:
y = x+4
y = x^2 − 2
(−2,2) and (3,7)
What does the range represent in an area model?
All possible output (area) values
Factor: 27x^6 − 8y^3
(3x^2 − 2y) (9x^4 + 6x^2y + 4y^2)
Find the domain of
f(x) = 1 / [(x+5)(x−3)]
x ≠ −5, 3
If f(x)=−3∣x+2∣−1 is shifted left 5 and down 4, write g(x).
g(x)=−3∣x+7∣−5
Solve:
y = −∣x−2∣ + 4
y + x = 2
(0,2)
Solve by completing the square:
x^2 + 12x + 40 = 0
x = −6 ± 2i
If x−2 is a factor of
x^3 − 2x^2 − x + 2
what is another factor?
x^2 − 1
What are the asymptotes of y = (x + 4) / (x−6)
y=1, x=6
How does reflecting f(x)=x^4 over the x‑axis change the equation?
It becomes −x^4
What does “select all viable solutions” mean?
Points that satisfy every inequality
How long until the object hits the ground?
h(t) = −16t^2 + 96t + 32
6 seconds
Find the zeros of (x+2)(x−4)(3x−5)
−2, 4 and 1.67 or 5/3
Translate y = 8 / x right 3 and up 4.
y = 8 / (x−3) + 4
Which transformations create h(x) = 5−x^4 from f(x) = x^4?
Reflect over x‑axis; translate up 5; reflect over y‑axis
For y > (x−1)^2 + 1 and y < x + 3
Which points are solutions?
(1,2) and (2,3)
Does the ball reach 20 m if
h(t) = −4t^2 + 12t + 15 and why?
Yes, because the equation has real solutions
Which statements about
f(x) = x^4 − x^3 + 3x^2 − 2x − 4 are true?
A The degree of the polynomial is 3.
B It is written in standard form
C As x increases or decreases, f(x) increases.
D The polynomial function has a maximum.
E The leading term determines the end behavior of the graph.
B - Standard form correct,
C - end behavior up/up, and
E - The leading term determines the end behavior of the graph.
Which equations have a horizontal asymptote at y = 2
A y = (2x+1) / (x−4)
B y = 5 / (x−2)
C y = (2x^2 − 3) / (x^2 + 1)
D y = (x + 2) / (2x + 5)
A. (2x+1) / (x−4) and
C. (2x^2 − 3) / (x^2 + 1)