Digital SAT · Solve it in Desmos

How to Solve Function Transformations on Desmos (Digital SAT)

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.

Perfect1600 Content Team·Uses the same Desmos calculator that is built into the Digital SAT

Full method for function transformations

The Desmos steps

1
Graph both functions
Type f(x) and the transformed version on separate lines in Desmos.
2
Watch the movement
See how the graph shifts, reflects, or stretches from the original.
3
Read a transformed point
Click a point to confirm where a specific point of the original moved.

Worked in Desmos

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.

Try it with the calculator

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.

Start the free diagnostic

Desmos questions

Can Desmos solve function transformations questions on the SAT?

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.

Is the Desmos calculator really built into the Digital SAT?

Yes. The Bluebook testing app includes the Desmos graphing calculator on every Math question, so the method here is one you can use on test day, not a workaround.

Should I still learn the algebra?

Yes. Desmos is fastest when you know what to type and what to read. Learn the underlying method on the full guide, then use the calculator to move quickly and to check your work.

Master function transformations, calculator and all

Every Digital SAT question type comes with the method, the Desmos shortcut, and free practice with instant feedback.

See the full method Take the free diagnostic

The Desmos steps

Graph both functions

Watch the movement

Read a transformed point

Worked in Desmos

Try it with the calculator

Question 3 is ready when you are

Desmos questions

Can Desmos solve function transformations questions on the SAT?

Is the Desmos calculator really built into the Digital SAT?

Should I still learn the algebra?

Master function transformations, calculator and all