Parent Functions & Transformations — Print
A handful of basic shapes, moved and stretched, account for almost every graph you will meet.
What the practice 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.
Trace a parabola onto tracing paper, then slide the paper.
Nothing about the curve changed. It is the same width, the same shape, the same everything.
All that changed is where it sits. Two numbers describe that: how far across, how far up.
Now pull the paper wider without moving it. Still the same curve, just stretched — one more number.
So a graph needs two things: which family it belongs to, and what was done to it. y = a(x-h)2 + k is exactly that sentence written in symbols.
The families worth knowing by sight
Six shapes cover most of the course: the line, the parabola x2, the cubic x3, the absolute value |x|, the root √x and the reciprocal 1x.
Learn what each one looks like at rest, and every later graph becomes one of them, moved.
Sketch each from three points if you are unsure — the shape returns faster than the formula does.
Why these six
They are the graphs you cannot build from the others. Everything else in Algebra II is one of these six with numbers attached.
Outside the function moves it vertically
Anything done outside f happens to the output, so it affects the graph vertically and does what it says.
f(x) + 3 lifts the graph three; -f(x) flips it upside down; 2f(x) stretches it to twice the height.
These are the intuitive ones, because the arithmetic happens after the function has already done its job.
Outside is honest
Add three outside, the graph rises three. What you write is what you get — vertically.
Inside the function moves it horizontally, backwards
Anything done inside the bracket happens to the input, and the graph responds in the opposite direction.
f(x - 3) moves the graph three to the right, not left. To get the output f used to give at 0, you must now feed in 3.
Likewise f(2x) squashes the graph to half its width, because the input reaches its usual values twice as fast.
The one everybody gets backwards
Ask what x makes the bracket zero. In f(x-3) that is x = 3, so the graph’s starting point has moved to 3. The bracket tells you the answer directly.
Reading a transformed formula
In y = a f(x - h) + k, read h and k as the new home of the parent point, and a as the stretch.
For y = -2(x+1)2 + 5: the vertex sits at (-1, 5), the graph is twice as tall and opens downward.
Order matters when you apply them by hand: stretch and flip first, then translate.
Check with one point
Put the parent’s key point through the transformation and see where it lands. If the vertex of (x+1)2 + 5 does not come out at (-1,5), a sign slipped.
Piecewise functions
Some situations need different rules on different stretches: a parking charge that is flat for two hours, then hourly.
Write each rule with the interval it governs, and evaluate by first asking which piece the input falls in.
On the graph, a filled dot means the endpoint is included; an open dot means it is not.
One input, one piece
The intervals must not overlap, or a single input would have two outputs — and then it is not a function at all.
Worked examples
Describing a transformation
Describe how y = -12(x - 4)2 + 3 relates to y = x2.
- The bracket is (x-4), so the graph moves 4 right. Inside the function, and the opposite of the sign shown.
- The -12 flips it downward and squashes it to half height. Outside the function, so vertical and honest.
- The +3 lifts it 3. The vertex lands at (4, 3). Outside again, so straight up.
Right 4, reflected down, half as tall, up 3; vertex (4,3)
Writing the formula from a picture
An absolute value graph has its corner at (-2, 1) and opens upward, twice as steep as y = |x|. Write its equation.
- Corner at x = -2 means the bracket is (x + 2). The bracket must be zero at the corner.
- Corner at y = 1 means +1 outside. Vertical shift is exactly the y-value.
- Twice as steep is a factor of 2 in front. Positive, since it still opens upward.
y = 2|x + 2| + 1
Evaluating a piecewise function
For f(x) = 2x + 1 when x < 3, and f(x) = x2 when x ≥ 3, find f(2) and f(3).
- 2 < 3, so f(2) uses the first rule: 2(2) + 1 = 5. Choose the piece before doing any arithmetic.
- 3 ≥ 3, so f(3) uses the second rule: 32 = 9. The ≥ puts the endpoint 3 in the second piece.
- The two rules disagree at 3 — the graph jumps. That is allowed; it is still one output per input.
f(2) = 5, f(3) = 9
Words to know
- Parent function
- The plainest member of a family — y = x2, y = |x|, y = √x, y = 1x — before anything is done to it.
- Translation
- A slide. f(x) + k moves the graph up; f(x - h) moves it right.
- Reflection
- A flip. -f(x) flips across the x-axis; f(-x) flips across the y-axis.
- Vertical stretch
- Multiplying the whole output: a f(x) pulls the graph away from the x-axis when |a| > 1.
- Piecewise function
- A function defined by different rules on different stretches of the domain.
Mistakes to watch for
Shifting horizontally the wrong way. f(x - 3) moves the graph right, not left. Ask which x makes the bracket zero: for (x-3) that is 3, so the graph now starts there.
Treating an inside stretch as a vertical one. f(2x) halves the width; 2f(x) doubles the height. They look alike in print and do opposite things, so check whether the 2 is inside the bracket.
Applying the shift before the stretch. In y = 2(x-1)2 + 3 the stretch acts on the parent, then the shift moves the result. Shifting first moves the wrong curve and lands the vertex in the wrong place.
Squaring the coefficient by accident. In y = (2x)2 the 2 is inside and gets squared to 4; in y = 2x2 it is outside and stays 2. The graphs differ by a factor of two in height.
Using the wrong piece at a boundary. If the definition says x ≥ 3, then 3 belongs to that piece. The inequality sign at the endpoint is the whole instruction, and a filled or open dot on the graph says the same thing.
Graphic designers call this the same thing mathematicians do: a transform. Moving, scaling and flipping a shape are the only three operations most design tools offer.
The reason y = f(x - 3) shifts the graph right and not left catches almost everyone. The graph does whatever it must so the function still sees its usual input.
Every animation you have watched is a transformation applied over and over, a fraction of a second apart.
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