Digital SAT · Nonlinear Graphs

How to Solve Function Transformations on the Digital SAT

A transformation slides or reshapes a graph. Change something outside the function and it moves vertically; change the input inside and it moves horizontally, opposite the sign. A negative in front reflects it, and a multiplier stretches or squeezes it. Read which kind of change the equation makes against the original, apply it to a point or the whole curve, or just graph both in Desmos.

Written from Perfect1600’s analysis of every function transformations question in our bank·Method checked against the current Bluebook test

per test: less than 1 per test
typical difficulty: Medium
practice questions: 21

How we counted

Frequency reflects how often this question type appears on a full-length Digital SAT. The difficulty mix reflects every question of this type across our bank.

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What the question bank shows

the rule

Outside vertical, inside reversed

Outside moves the graph up or down; inside moves it horizontally, opposite the sign.

33%

Check on Desmos

Graph the original and the new function to see the movement.

the hazard

Horizontal shift

f(x minus 2) shifts right, which is the reversal students miss.

How to recognize function transformations questions

A function is given, then altered, like f(x) plus 3 or f(x minus 2).
You are asked for the new graph, a shifted point, or how the graph changed.
The words shifted, reflected, translated, or stretched appear.
Choices are graphs, points, or transformed equations.

Why students miss these

The inside change runs against intuition: f(x minus 2) shifts right, toward the positive direction, even though it reads like subtraction. People shift the wrong way, mix up inside and outside changes, or miss a reflection buried in a sign. Keep outside changes vertical and inside changes horizontal and reversed, and the direction stays straight.

The step-by-step method

1
Compare to the original
See how the new equation differs from f(x): a change outside, inside, or a sign or multiplier.
2
Outside moves vertically
Adding outside lifts the graph up; subtracting drops it down.
3
Inside moves horizontally, reversed
f(x minus h) shifts right by h; f(x plus h) shifts left, opposite the sign.
4
Reflections and stretches
A negative in front reflects over the x-axis; a multiplier stretches or squeezes.

Solving it on Desmos

Graph both functions. Type f(x) and the transformed version on separate lines in Desmos.
Watch the movement. See how the graph shifts, reflects, or stretches from the original.
Read a transformed point. Click a point to confirm where a specific point of the original moved.

Full Desmos walkthrough for function transformations→

When to use it: Graphing both functions shows the transformation outright, the quickest fix for the reversed horizontal shift. Reason from the equation when only a single transformed point is asked.

Worked examples

Easy example

The function

f

is defined by

f(x) = x^{2}

, and

g

is defined by

g(x) = f(x - 3)

. Compared to the graph of

f

, the graph of

g

is shifted in which way?

left 3 units

right 3 units

up 3 units

down 3 units

A: Incorrect. A left shift comes from $f(x + 3)$ , not $f(x - 3)$ .

B: Correct. Replacing $x$ with $x - 3$ shifts the graph right 3 units.

C: Incorrect. A vertical shift comes from a change outside the function.

D: Incorrect. A downward shift comes from subtracting outside the function.

Explanation

Replacing $x$ with $x - 3$ shifts the graph right 3 units.

Medium example

The function

f

is defined by

f(x) = x^{2}

, and

g

is defined by

g(x) = -f(x)

. Which of the following describes the graph of

g

compared to the graph of

f

The graph is shifted down 1 unit.

A leftward shift of 1 unit occurs.

It is reflected across the

x

-axis.

Reflection across the

y

-axis occurs.

A: Incorrect. Negating the output is a reflection, not a vertical shift.

B: Incorrect. A horizontal shift comes from a change inside the function.

C: Correct. Multiplying the output by $-1$ , $g(x) = -f(x)$ , reflects the graph across the $x$ -axis.

D: Incorrect. A reflection across the $y$ -axis comes from $f(-x)$ .

Explanation

$g(x) = -f(x)$ negates outputs, reflecting across the $x$ -axis.

Hard example

The graph of

y = f(x)

passes through the point

(4, 10)

. The function

g

is defined by

g(x) = 3f(x)

. Which point must lie on the graph of

g

(12, 10)

(4, 30)

(4, 13)

(12, 30)

A: Incorrect. The factor 3 multiplies the output, not the input.

B: Correct. $g(4) = 3 f(4) = 3(10) = 30$ , so $(4, 30)$ lies on $g$ .

C: Incorrect. The factor 3 multiplies the output; it does not add 3.

D: Incorrect. The input is unchanged; only the output is scaled.

Explanation

Scaling outputs by 3: $g(4) = 30$ , giving $(4, 30)$ .

Detailed explanation

Multiplying by 3 outside, $g(x) = 3f(x)$ , is a vertical stretch that scales each output by 3 while leaving inputs fixed. So $g(4) = 3(10) = 30$ and the point is $(4, 30)$ . Scaling the input or adding 3 gives the other points.

The common traps

Pattern	What it does	The tell
Horizontal shift reversed	Shifted left for f(x minus h) instead of right.	Inside changes move opposite the sign: minus h goes right.
Inside versus outside	Applied an inside change vertically, or an outside change horizontally.	Outside the function is vertical; inside is horizontal.
Reflection missed	Ignored a negative sign that reflects the graph.	A negative in front flips the graph over the x-axis.

Try it: two real questions

Question 1easy

The graph of a function is shown. The graph will be translated up 3 units. Which of the following is the graph of the resulting function?

Question 2easy

In the $xy$ -plane, the graph of $g(x) = (x + 2)^{2}$ is a parabola. What is the vertex of this parabola?

Question 3 is ready when you are

Keep going with more questions of this type, each with the same step-by-step reasoning. Free to start.

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Often confused with

Quadratic & Exponential Functions

Quadratic Equations

Common questions

How do function transformations work on the SAT?

Outside changes move the graph vertically: plus up, minus down. Inside changes move it horizontally and opposite the sign: f(x minus 2) shifts right. A negative reflects it, a multiplier stretches it.

Why does f(x minus 2) shift right instead of left?

Because the input reaches the same value 2 units later, so the whole graph slides right by 2. Inside changes move opposite the sign they show.

How do I reflect a function's graph?

A negative in front, as in negative f(x), reflects over the x-axis. Replacing x with negative x reflects over the y-axis instead.

Can Desmos show a transformation?

Yes. Graph the original f(x) and the transformed function together, and the shift, reflection, or stretch is visible, which fixes the reversed horizontal shift.

Practice function transformations the way it is tested

Start with the free 16-question diagnostic, then drill this type with step-by-step reasoning on every question.

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What the question bank shows

How to recognize function transformations questions

Why students miss these

The step-by-step method

Compare to the original

Outside moves vertically

Inside moves horizontally, reversed

Reflections and stretches

Solving it on Desmos

Worked examples

The common traps

Try it: two real questions

Question 3 is ready when you are

Often confused with

Related reading

Common questions

How do function transformations work on the SAT?

Why does f(x minus 2) shift right instead of left?

How do I reflect a function's graph?

Can Desmos show a transformation?

Practice function transformations the way it is tested