Using Transformations to Graph Functions

Step 2 of 4

Learn

The general form, the point rule, three worked examples and the sequence builder.

Real-world idea

One editing app, four sliders

Imagine an editing app with four sliders for a photo: size, flip, sideways position and up/down position. Any combination of settings is possible — and you can always undo them one at a time.

Every transformation of a parent graph works the same way. The four numbers a, k, d and c are the sliders, and g(x) = af(k(x − d)) + c is the full set of controls.

Key term

General transformation form

g(x) = a·f(k(x − d)) + c

g(x) = 2√(3(x − 1)) + 5 uses f(x) = √x with a = 2, k = 3, d = 1, c = 5.

a — vertical scaling by |a|; a vertical stretch if |a| > 1, a vertical compression if 0 < |a| < 1, and a reflection in the x-axis if a < 0.

k — horizontal scaling by 1/|k|; a horizontal compression if |k| > 1, a horizontal stretch if 0 < |k| < 1, and a reflection in the y-axis if k < 0.

d — horizontal translation: right if positive, left if negative.

c — vertical translation: up if positive, down if negative.

Point mapping

Every point (x, y) on f moves to (x/k + d, ay + c) on g, where k ≠ 0. Do the scaling and reflections first, then the translations — this is the order the mapping uses, so it is the order your graph follows too.

Try it

Rewrite f(4x − 12) so you can read k and d directly.

af(4(x − 3))bf(4(x − 12))cf(4(x + 3))df(4(x + 12))

Try it

The point (6, 5) is on f(x). Where does it land on g(x) = −2f(3(x − 2)) + 4?

a(4, −6)b(4, 14)c(0, −6)d(20, −6)

Worked example

Example A — Graph g(x) = −2√(2(x − 3)) + 1 using f(x) = √x.

Read the values
Match g to a·f(k(x − d)) + c.
a = −2, k = 2, d = 3, c = 1
Describe the transformations
Horizontal compression by 1/2, vertical stretch by 2, reflection in the x-axis, 3 right, 1 up.
Map key points
Use (x, y) → (x/2 + 3, −2y + 1).
(0, 0) → (3, 1); (1, 1) → (3.5, −1); (4, 2) → (5, −3)
State domain and range
The curve starts at (3, 1) and goes down to the right.
Domain: x ≥ 3. Range: y ≤ 1

Answer: Starting point (3, 1); the curve runs down and to the right. Domain x ≥ 3; range y ≤ 1.

Example A — parent (grey) and g(x) = −2√(2(x − 3)) + 1

g(x) = −2√(2(x − 3)) + 1

Domain: x ≥ 3 · Range: y ≤ 1

Worked example

Example B — Graph g(x) = 3|0.5(x + 2)| − 4 using f(x) = |x|.

Read the values
x + 2 = x − (−2).
a = 3, k = 0.5, d = −2, c = −4
Describe the transformations
Horizontal stretch by 2, vertical stretch by 3, 2 left, 4 down.
Find the vertex
The vertex (0, 0) moves to (d, c).
(0, 0) → (−2, −4)
State domain and range
The V opens up from (−2, −4).
Domain: all real numbers. Range: y ≥ −4

Answer: Vertex (−2, −4); the V opens up and is 3 times as steep. Domain all real numbers; range y ≥ −4.

Example B — parent (grey) and g(x) = 3|0.5(x + 2)| − 4

g(x) = 3|0.5(x + 2)| − 4

Domain: all real numbers · Range: y ≥ −4

Same graph, different sliders

Different combinations of horizontal and vertical scaling can sometimes produce the same final picture. What matters is the graph you end up with — the sequence builder below will accept any correct order, as long as stretches and reflections come before translations.

Worked example

Example C — The equation g(x) = f(3x + 12) hides its shift. Rewrite it in transformation form.

Factor the coefficient of x
Take 3 out of both terms inside the bracket.
3x + 12 = 3(x + 4)
Rewrite
Now the form shows k and d directly.
g(x) = f(3(x + 4))
Read k and d
k = 3, and x + 4 = x − (−4).
k = 3, d = −4
Describe
The graph is compressed horizontally by 1/3 and shifted 4 units left.
Shift: 4 left — not 12 left

Answer: g(x) = f(3(x + 4)): compression by 1/3 and a shift 4 left. The shift is 4, not 12.

Transformation Sequence Builder

Identify a, k, d and c, then build the graph step by step — or find the mistake.

Target · parent f(x) = √x

g(x) = −2√(2(x − 3)) + 1

Step 1 · Identify a, k, d and c

Scale and flip first, slide last.

Every point (x, y) moves to (x/k + d, ay + c).

Using Transformations to Graph Functions

Warm Up

Learn

One editing app, four sliders

General transformation form

Practice

Build it

Apply it

Challenge

Explain it

Mastery Check

Lesson 1.8 mastery check