Unit 5 · FunctionsF4 · Transformations · Guidance

Function Transformations

Read shifts, stretches, and reflections from an equation before graphing.

30 min
Unlocks: Course Correction
Desmos way

Ready when you are

7 steps · about 30 minutes

Start with the warm-up.

Start

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Warm-up

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Key terms

horizontal shift: a left-right movement caused inside the input

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Try it first

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Try this before the explanation. Commit to a setup, even if you are unsure.

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The big idea

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The decision that matters

Outside changes act vertically; inside changes act horizontally in the opposite direction. Write that decision before calculating. It turns a crowded ACT stem into a familiar structure and makes the likely distractors easier to spot.

Key idea

Structure before arithmetic

Outside changes act vertically; inside changes act horizontally in the opposite direction. Keep exact values through the setup; round only when the stem asks.

Core relationship

g(x)=a f(b(x-h))+k

Use the formula only after matching every symbol to the diagram or equation.

Algebra or Desmos?

Read horizontal changes from inside the input and vertical changes from outside the function. Desmos is useful for checking the predicted vertex and reflection after you state those transformations.

Explore in Desmos

Match equations to transformed vertices

Click each vertex and match its coordinates to the shifts in the equation.

The calculator is live: change anything and see what happens.

Match equations to transformed vertices

Degree mode

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Trap

The main trap

The graph of

f(x+4)

moves left 4, because the input reaches the old value 4 units earlier.

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Worked examples

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Skip check · 2 questions

Already know this? Get both right on the first try and you can skip the worked examples.

Worked example

Example 1

Problem

Starting from

y=x^2

, describe the transformations and find the vertex of

g(x)=(x-3)^2+2

Start from

y=x^2

x-3

shifts right 3.

Which point is the new vertex?

The outside

+2

shifts up 2; the vertex is

(3,2)

Why does the inside subtraction move the graph right?

Worked example

Example 2: ACT twist

Problem

Describe the transformations of

y=|x|

that produce

h(x)=-2|x+1|+5

, and state the vertex.

x+1

shifts left 1.

What effect does the outside factor $-2$ have?

The vertex is

(-1,5)

Why does the negative multiplier affect outputs rather than inputs?

Faded example · you finish it

Example 3: You finish it

Problem

Describe

y=\frac12f(x)-6

Vertical distances are halved.

After the vertical compression by $\frac12$ , identify the remaining transformation.

Desmos way

Verify two transformed vertices

Click each vertex and match its coordinates to the shifts in the equation.

Verify two transformed vertices

Degree mode

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Practice

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Spot the mistake

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Spot the mistake

Where did it go wrong?

One line has an error. Click the line where it first goes wrong.

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Exit ticket

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Optional · 4 questions · Hardest levelChallenge set: the kind of question that decides a 36

These are last-ten-questions hard. Take your time, use Desmos when it helps, and expect to miss some: that's where the learning is.