Move and reshape a function · Spellman Computer Science

Worked example

Do Now · Easy

Do this: Try this on paper, then compare with the worked check.

Key words

Key word · Unit 6TransformationA change that moves, stretches or reflects a graph.Exactly: A mapping of a function graph, such as $g (x) = a f (b (x - h)) + k$ , with $a \neq 0$ and $b \neq 0$ .Mix-up alert: An inside shift x - h moves the graph right by h when h is positive.See this in Key words

Map the parent vertex $(0, 0)$ to $y = (x - 2)^{2} + 3$ . Check: the new vertex is $(2, 3)$ .

Big ideas

The ideas you will use

Do this: Read one idea, then explain it in your own words.

Outside changes outputs
In $g (x) = a f (x) + k$ , multiplication changes output scale and adding k moves the graph vertically. A negative a also reflects across the x-axis.
Inside changes inputs
In $f (x - h)$ , a horizontal shift moves the graph right by h. Solve the inside input condition to locate a familiar point instead of guessing from a sign.
Horizontal scale is reciprocal
In $f (b x)$ , with b nonzero, the horizontal scale is reciprocal: each original x-coordinate becomes $\frac{x}{b}$ . A factor with magnitude larger than 1 compresses horizontal distances.
Track a known point
For $g (x) = a f (b (x - h)) + k$ , with a and b nonzero, a known point $(u, v)$ becomes $(h + \frac{u}{b}, a v + k)$ . Factor the inside expression before reading h. Check a second point and the new domain and range.

Worked example

Guided practice · Medium

Do this: Cover each worked step and try it on paper before reading on.

Map $(8, 3)$ on f to $g (x) = - 2 f (4 (x + 1)) + 5$ . Solve $4 (x + 1) = 8$ for $x = 1$ . Change the output to $- 2 (3) + 5 = - 1$ . The image is $(1, - 1)$ ; horizontal and vertical changes act on different coordinates.

Step-through

A problem-solving desk

Do this: Step through the example. Predict the next step before revealing it.

Worked example: locate a vertex

4 steps. Tap a number or Show me how it works.

Worked example: locate a vertex

Do this: Cover the next step and try it yourself.

Step 1: Let $g (x) = - 2 {(x - 3)}^{2} + 1$ . Start with the parent $f (x) = x^{2}$ .
Step 2: The inside input is zero at $x = 3$ , so the vertex moves to $x = 3$ .
Step 3: The outside scale is −2 and vertical shift is +1; the vertex is $(3, 1)$ and the graph opens downward.
Step 4: $g (4) = - 1$ , below the vertex.

The range is $(- \infty, 1]$ .

Worked example: track a point

Do this: Cover the next step and try it yourself.

Step 1: Suppose $(6, 4)$ is on $y = f (x)$ . Find its corresponding point on $y = f (2 x) + 3$ .
Step 2: The input condition is $2 x = 6$ , so the new x-coordinate is 3.
Step 3: The output becomes $4 + 3 = 7$ , giving point $(3, 7)$ .
Step 4: Substituting $x = 3$ uses $f (6) + 3 = 7$ .

Horizontal coordinates were divided by 2, not multiplied.

Mix-up alert: check the tempting shortcut

Do this: Explain why the tempting shortcut fails.

Wrong idea: $f (x - 3)$ moves left because it contains a minus sign. The original input 0 occurs when $x - 3 = 0$ , so $x = 3$ . Use that equation to see a right shift.

Watch

Video: Move and reshape a function

Do this: Watch the public lesson, pause to try a step, then return to your practice.

Watch on YouTubeOpens in a new tab. Keep your work here.

Watch for:

Factoring the input before reading horizontal transformations.
Pause before a worked step and try it on paper. Compare the reasoning, not just the final answer.

Open on YouTube

From The Organic Chemistry Tutor · Transformations of Functions | Precalculus

The linked video is free to watch. The creator may offer optional paid resources; they are not needed for this lesson. If YouTube is blocked, use the lesson notes and worked examples here.

Step 1 of 4: Learn it

Do Now · Easy

Key words

The ideas you will use

Outside changes outputs

Inside changes inputs

Horizontal scale is reciprocal

Track a known point

Guided practice · Medium

A problem-solving desk

Worked example: locate a vertex

Worked example: track a point

Mix-up alert: check the tempting shortcut

Video: Move and reshape a function

Watch for:

Step 2 of 4: Practice it

Eight ideas to recall

Question

Answer

Question

Answer

Question

Answer

Question

Answer

Question

Answer

Question

Answer

Question

Answer

Question

Answer

Test the boundaries

Statement 1 of 4

Check the foundations

Apply and explain

Explain your reasoning

Explain how to sketch g(x)=−(x−2)2+3 from y=x2. Give the vertex and one checking point.

Optional challenge · Hard

Let g(x)=−2f(3x−6)+5. The graph of f has domain [−3,6] and range [−1,4]. State the transformed intervals and map (3,2).

Match the key words

Word match round

Terms

Meanings

Step 3 of 4: Homework

Move and reshape a function practice preview

Step 4 of 4: Finish

Ready to move on?

Step 1 of 4: Learn it

Do Now · Easy

Key words

The ideas you will use

Outside changes outputs

Inside changes inputs

Horizontal scale is reciprocal

Track a known point

Guided practice · Medium

A problem-solving desk

Worked example: locate a vertex

Worked example: track a point

Mix-up alert: check the tempting shortcut

Video: Move and reshape a function

Watch for:

Step 2 of 4: Practice it

Eight ideas to recall

Question

Answer

Question

Answer

Question

Answer

Question

Answer

Question

Answer

Question

Answer

Question

Answer

Explain how to sketch $g (x) = - {(x - 2)}^{2} + 3$ from $y = x^{2}$ . Give the vertex and one checking point.

Let $g (x) = - 2 f (3 x - 6) + 5$ . The graph of f has domain $[- 3, 6]$ and range $[- 1, 4]$ . State the transformed intervals and map $(3, 2)$ .

Explain how to sketch $g (x) = - {(x - 2)}^{2} + 3$ from $y = x^{2}$ . Give the vertex and one checking point.

Let $g (x) = - 2 f (3 x - 6) + 5$ . The graph of f has domain $[- 3, 6]$ and range $[- 1, 4]$ . State the transformed intervals and map $(3, 2)$ .