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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. 1

    Graph both functions

    Type f(x) and the transformed version on separate lines in Desmos.

  2. 2

    Watch the movement

    See how the graph shifts, reflects, or stretches from the original.

  3. 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 ff is defined by f(x)=x2f(x) = x^{2}, and gg is defined by g(x)=f(x−3)g(x) = f(x - 3). Compared to the graph of ff, the graph of gg is shifted in which way?
A
left 3 units
right 3 units
C
up 3 units
D
down 3 units

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

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

Medium example
The function ff is defined by f(x)=x2f(x) = x^{2}, and gg is defined by g(x)=−f(x)g(x) = -f(x). Which of the following describes the graph of gg compared to the graph of ff?
A
The graph is shifted down 1 unit.
B
A leftward shift of 1 unit occurs.
It is reflected across the xx-axis.
D
Reflection across the yy-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 −1-1, g(x)=−f(x)g(x) = -f(x), reflects the graph across the xx-axis.

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

Explanation

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

Hard example
The graph of y=f(x)y = f(x) passes through the point (4,10)(4, 10). The function gg is defined by g(x)=3f(x)g(x) = 3f(x). Which point must lie on the graph of gg?
A
(12,10)(12, 10)
(4,30)(4, 30)
C
(4,13)(4, 13)
D
(12,30)(12, 30)

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

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

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)=30g(4) = 30, giving (4,30)(4, 30).

Detailed explanation

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

Try it with the calculator

Question 1easy
-3-2-1123-4-224681012Oxy

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 xyxy-plane, the graph of g(x)=(x+2)2g(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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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.

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