Knowledge Notes | Carey AP Precalculus Practice Lab

Follow point-coordinate changes in order: right 3, vertical factor 2, then up 1. — Follow point-coordinate changes in order: right $3$ , vertical factor $2$ , then up $1$ .

What a strong response shows

Factor the input before reading a horizontal shift.
Inside changes alter the input; outside changes alter the output.
Use one known point as a verification.

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What You Need to Know

Key idea

A transformed function changes the inputs, the outputs, or both. The outside changes affect $y$ ; the inside changes affect $x$ .

Fast decision rule

Read $g\left(x\right)=a$ $f\left(b \left(x-h\right)\right)+k$ from the inside out for the horizontal move, then apply the vertical scale or reflection and the vertical shift.

Worked example

In $g\left(x\right)=2 f\left(x-3\right)+4$ , the graph moves right $3$ , doubles its output distance from the x-axis, and moves up $4$ .

Common mistake

Reading $x-3$ as a shift left $3$ . Inside signs act opposite to the visible horizontal direction.

Mini check

What visible change is caused by adding $5$ outside the function?

Learning Objectives

Key idea

1.12.A asks you to construct additive and multiplicative transformations and explain how they change a representation.

Fast decision rule

Translate the verbal change into input and output rules before writing the final formula.

Worked steps

$1$ . Identify the original function. $2$ . Write the horizontal input rule. $3$ . Write the vertical output rule. $4$ . Substitute both into the function. $5$ . Test one known point.

Common mistake

Combining several transformations while skipping the order or reciprocal horizontal factor.

Mini check

Which part of $g\left(x\right)=a$ $f\left(b \left(x-h\right)\right)+k$ controls the vertical reflection?

Key Concepts

Key idea

Translations move a graph, dilations change scale, and negative multipliers reflect a graph. Horizontal factors use a reciprocal scale.

Fast decision rule

Outside: $y^{\prime}=a$ $y+k$ . Inside: $x^{\prime}=h+\frac{x}{b}$ . A negative a flips outputs; a negative $b$ reverses inputs.

Worked example

If $f$ contains the point $\left(3,\,2\right)$ and $g\left(x\right)=-2 f\left(3 \left(x-1\right)\right)+4$ , then $x^{\prime}=1+\frac{3}{3}=2$ and $y^{\prime}=-2 \left(2\right)+4=0$ .

Common mistake

Multiplying the x-coordinate by $b$ instead of solving b(x−h)=x_original.

Mini check

If $b=4$ , is the horizontal scale factor $4$ or $\frac{1}{4}$ ?

Relevant Representations

Key idea

The same transformation can be read from a formula, a graph, a table, or a set of mapped points.

Fast decision rule

Use a point or table to verify a formula; use a graph to check direction, scale, and visible features.

Read the transformation from the representation
Representation	Look for	Check
Formula	inside and outside operations	signs, reciprocal factor, and order
Graph	shift, reflection, width, and height	one mapped point and one characteristic
Table	input and output pairs	the transformed input and output rules
Domain/range	allowed inputs and outputs	reversal when a multiplier is negative

Common mistake

Matching only the shape while ignoring the location of a key point or endpoint.

Mini check

Which representation makes a domain change easiest to verify: a graph, a table, or a formula?

Essential Relationships or Formulas

Key idea

A compact transformation model keeps the input and output rules visible.

g(x)=a\,f\!\bigl(b(x-h)\bigr)+k

x_{\text{new}}=h+\frac{x_{\text{old}}}{b},\qquad y_{\text{new}}=a\,y_{\text{old}}+k

Fast decision rule

Solve the inside equation for the new input; apply the outside rule directly to the old output.

Worked example

For $g(x)=-\tfrac12 f(2(x+3))+1$ and $f(4)=6$ , the preimage input satisfies $2(x+3)=4$ , so $x=-1$ ; the new output is $-\tfrac12(6)+1=-2$ .

Common mistake

Applying the vertical multiplier to $x$ or using $b$ instead of $\frac{1}{b}$ for horizontal scale.

Mini check

If $f$ has domain $\left[-2,\,6\right]$ and $g\left(x\right)=f\left(2 x-4\right)$ , what is the domain of $g$ ?

Worked Examples

Key idea

Build a transformation by mapping one known point first, then check the whole graph or table.

Fast decision rule

If the prompt gives a sequence, preserve the sequence; if it gives a final formula, read the completed inside and outside rules.

Worked steps

To shift a graph left $2$ and down $3$ , replace $x$ with $x+2$ and subtract $3$ outside: $g\left(x\right)=f\left(x+2\right)-3$ .

Worked steps

To compress horizontally by $\frac{1}{2}$ , use $f\left(2 x\right)$ , because the input reaches the old x-value twice as fast.

Worked steps

To map a domain, solve the inside input equation at both endpoints; to map a range, apply the output rule and reverse endpoint order when $a<0$ .

Common mistake

Describing $f\left(2 x\right)$ as a stretch by $2$ or translating before checking the written order.

Mini check

Write a formula for a vertical reflection, a shift right $4$ , and a shift up $2$ .

Common Mistakes

Key idea

Transformation errors are usually sign errors, reciprocal errors, or order errors.

Fast decision rule

Test one easy original point and one endpoint before accepting a transformed graph.

Worked example

If a point $\left(-2,\,5\right)$ is transformed by $g\left(x\right)=-3 f\left(x+1\right)+2$ , the new input solves $x+1=-2$ , so $x=-3$ ; the new output is $-13$ .

Common mistake

Saying “inside plus $1$ moves right” or forgetting that a negative output multiplier reverses the range.

Mini check

In $g\left(x\right)=f\left(-x\right)$ , which axis supplies the reflection?

AP Exam Tips

Key idea

A correct transformation answer must match both the algebra and at least one visible feature.

Fast decision rule

Check inside sign, reciprocal factor, outside sign, vertical shift, and one mapped point in that order.

Worked steps

$1$ . Write the input rule. $2$ . Solve for the new x-coordinate. $3$ . Write the output rule. $4$ . Map a point or endpoint. $5$ . Check domain and range.

Common mistake

Selecting a graph with the right shape but the wrong location, width, or endpoint behavior.

Mini check

Can you justify your answer with one mapped point and one domain or range statement?

Practice

Use the Questionbank items below after checking the input and output rules.

TRY IT IN A NEW SETTING

Can you transfer the method?

The point $\left(-2,\,5\right)$ lies on $f$ . Find its transformed point on $g(x)=-3f(x+1)+2$ .

Compare your reasoning

Solve $x+1=-2$ , so the new input is $-3$ . The new output is $-3\cdot 5+2=-13$ . The transformed point is $\left(-3,\,-13\right)$ .

Use it in a real question

Work independently, then compare the mathematical evidence in your answer.

Practise this topic Check the whole unit

1.12 · Transformations of Functions

Topic 1.12 · Transformations of Functions

What a strong response shows

What You Need to Know

Learning Objectives

Key Concepts

Relevant Representations

Essential Relationships or Formulas

Worked Examples

Common Mistakes

AP Exam Tips

Can you transfer the method?