Parent Functions & Transformations — Practice
A handful of basic shapes, moved and stretched, account for almost every graph you will meet.
What these questions cover
These questions use 8 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Absolute value — |x| — distance from zero, always positive or zero.
- Open — Open excludes the endpoint (<, >); closed includes it (≤, ≥).
- Vertex — A flat surface; a line where two faces meet; a corner point.
- Square root — For k ≥ 0, √k is the nonnegative number whose square is k.
- Factor — A number that divides evenly into another.
- Function — A rule pairing each input with exactly one output.
- Parabola — The U-shaped graph of a quadratic function.
- Height — The perpendicular distance from the base to the opposite corner.
Problem 1
How does y=x2+5 compare with y=x2?
- Up 5
- Down 5
- Right 5
- Left 5
Worked answer
Up 5
Outside means vertical
Adding lifts the graph
Problem 2
How does y=(x-3)2 compare with y=x2?
- Right 3
- Left 3
- Up 3
- Down 3
Worked answer
Right 3
x-3=0 at x=3
So the vertex has moved to 3 — right
Problem 3
What does y=-x2 do to y=x2?
- Flips it across the x-axis
- Flips it across the y-axis
- Shifts it down
- Nothing
Worked answer
Flips it across the x-axis
Every output changes sign
So the graph turns upside down
Problem 4
Give the vertex of y=(x-2)2+7.
Worked answer
(2,7)
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Problem 5
Give the vertex of y=(x+4)2-1.
Worked answer
(-4,-1)
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Problem 6
Give the vertex of y=-3(x-1)2+8.
Worked answer
(1,8)
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Problem 7
By what factor is y=4|x| taller than y=|x|?
Worked answer
4
Every height is multiplied by 4
Problem 8
What does y=f(2x) do to the graph of y=f(x)?
- Halves its width
- Doubles its width
- Doubles its height
- Halves its height
Worked answer
Halves its width
The input reaches its usual values twice as fast
So the graph is squashed to half the width
Problem 9
An absolute value graph has its corner at (3,-2) and is not stretched. Write its equation.
Worked answer
y=|x-3|-2
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Problem 10
A parabola has vertex (-1,4) and opens upward, same width as y=x2. Write its equation.
Worked answer
y=(x+1)^2+4
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Problem 11
For f(x)=x+2 when x<1 and f(x)=x2 when x ≥ 1, find f(0).
Worked answer
2
0<1, so use x+2
0+2=2
Problem 12
For the same rule, find f(1).
Worked answer
1
1 ≥ 1, so use x2
12=1
Problem 13
For y=-2(x+3)2+1, give the vertex.
Worked answer
(-3,1)
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Problem 14
Which graph is y=√x+4 compared with y=√x?
- Left 4
- Right 4
- Up 4
- Down 4
Worked answer
Left 4
x+4=0 at x=-4
The graph now starts at -4, which is 4 to the left
Problem 15
At what x does y=√x-6 begin?
Worked answer
6
x-6 ≥ 0
x ≥ 6
Problem 16
For f(x)=3 when x<0 and f(x)=x+3 when x ≥ 0, find f(-5)+f(5).
Worked answer
11
f(-5)=3
f(5)=8
3+8=11