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Amy Ferguson Moncure

Algebra 1 · Unit 8

Graph Quadratics with Transformations

Graph f(x) + k and k f(x) for rational k, including shifts, stretches, and reflections.

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Key Ideas Easy Medium Hard Super Challenge
What to Remember

f(x) + k

Add k to every output. The graph shifts up if k > 0 and down if k < 0. The vertex’s y-value changes by k.

k f(x)

Multiply every output by k. |k| > 1 stretches away from the x-axis (narrower). 0 < |k| < 1 compresses (wider). k < 0 reflects over the x-axis.

k is rational here

This page uses values such as 2, 12, −1, and 32.

Graph in the answer

When you are asked to graph, the sketch is in the Show Answer.

Level 1

Easy

Integer k. Start from y = x².

Example
Graph g(x) = x² + 3. Compare to y = x².

f(x) + 3 shifts y = x² up 3.

Vertex (0, 3). Opens up.

Answer: up 3; vertex (0, 3)

1
Graph g(x) = x² − 4.
Show Answer

Shift y = x² down 4.

Vertex (0, −4).

Answer: down 4; vertex (0, −4)

2
Graph g(x) = 2x².
Show Answer

k = 2 stretches y = x² away from the x-axis. Narrower.

Vertex still (0, 0).

Answer: narrower; vertex (0, 0)

3
Graph g(x) = −1x².
Show Answer

k = −1 reflects y = x² over the x-axis.

Opens down. Vertex (0, 0).

Answer: opens down; vertex (0, 0)

4
How does g(x) = x² + 1 compare to y = x²?
Show Answer

Every point is 1 higher. Vertex (0, 1).

Answer: up 1

5
How does g(x) = 12x² compare to y = x²?
Show Answer

|k| = 12 < 1, so wider (compressed toward the x-axis).

Answer: wider

Level 2

Medium

A shift and a stretch, or k as a fraction.

Example
Graph g(x) = 2x² − 3. Compare to y = x².

First 2x² is narrower than x².

Then −3 shifts down 3.

Vertex (0, −3). Opens up.

Answer: narrower, down 3; vertex (0, −3)

1
Graph g(x) = 12x² + 2.
Show Answer

Wider than x², then up 2.

Vertex (0, 2).

Answer: wider, up 2; vertex (0, 2)

2
Graph g(x) = −2x² + 4.
Show Answer

Reflect and stretch, then up 4.

Vertex (0, 4). Opens down.

Answer: opens down, vertex (0, 4)

3
f(x) = x². Write g(x) = f(x) − 5 and sketch g.
Show Answer

g(x) = x² − 5

Answer: g(x) = x² − 5

4
f(x) = x². Write h(x) = 32f(x) and say whether h is narrower or wider than f.
Show Answer

32 > 1, so narrower.

h(x) = 32x²

Answer: narrower

5
The graph of y = x² is shifted down 1 and stretched by 3. Write g(x) and name the vertex.
Show Answer

g(x) = 3x² − 1

Vertex (0, −1)

Answer: g(x) = 3x² − 1; (0, −1)

Level 3

Hard

Start from a shifted parabola, then apply k f(x) or f(x) + k.

Example
f(x) = (x − 2)². Graph g(x) = f(x) + 3 and h(x) = 12f(x).

g(x) = (x − 2)² + 3. Vertex (2, 3).

h(x) = 12(x − 2)². Same vertex x = 2, but y = 0, and wider.

Answer: g vertex (2, 3); h vertex (2, 0), wider

1
f(x) = (x + 1)² − 4. Graph f(x) + 2.
Show Answer

Add 2 to k: vertex (−1, −2).

Answer: vertex (−1, −2)

2
f(x) = (x − 1)². Graph −1f(x).
Show Answer

Reflect over the x-axis. Vertex still (1, 0), opens down.

Answer: opens down; vertex (1, 0)

3
g(x) = −12x² + 3. Name the transformations from y = x² and sketch.
Show Answer

Wider, reflected, then up 3.

Vertex (0, 3). Opens down.

Answer: wider, flip, up 3

4
A student graphs 2x² + 4 by stretching first, then shifting, versus shifting first, then stretching. Which order matches 2x² + 4?
Show Answer

2(x² + 4) would stretch the shift and move the vertex to (0, 8).

2x² + 4 stretches y = x², then adds 4. Vertex (0, 4).

Add 4 last.

Answer: stretch, then add 4

5
f(x) = x² + 1. Write 3f(x) and f(x) − 12. Compare vertices.
Show Answer

3f(x) = 3x² + 3, vertex (0, 3)

f(x) − 12 = x² + 12, vertex (0, 12)

Answer: (0, 3) vs (0, 1/2)

Super Challenge

Super Challenge

1
k f(x) + c versus k(f(x) + c) for f(x) = x², k = 2, c = 3. Expand both and compare vertices.
Show Answer

2x² + 3 has vertex (0, 3).

2(x² + 3) = 2x² + 6 has vertex (0, 6).

The 3 is stretched only in the second form.

Answer: vertices (0, 3) and (0, 6)

2
A graph of y = x² is reflected and then shifted up 5. Write two functions that match, one with k f(x) and one in vertex form.
Show Answer

g(x) = −1x² + 5

Same as −1(x − 0)² + 5

Answer: g(x) = −x² + 5

3
Why does multiplying by 13 make a parabola look wider, even though every y-value got smaller in size?
Show Answer

Points are pulled toward the x-axis, so the same x now has a smaller |y|.

To reach y = 1 you need a larger |x| than before, which looks wider.

Answer: pulled toward the x-axis

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