Precalculus Transformations of Graphs Worksheet (PDF)

Precalculus math worksheet: Transformations of Graphs – Maths PDF

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This precalculus transformations worksheet for Precalculus covers transformations of graphs (Common Core F-BF.B.3, F-IF.C.7b), with a free PDF to print.

These 20 problems practice transformations of graphs (Common Core F-BF.B.3 and F-IF.C.7b). You will describe shifts, reflections and stretches, move points, write equations from graphs and test for even and odd functions.

Precalculus · Common Core State Standards: F-BF.B.3, F-IF.C.7b

Precalculus transformations worksheet: exercises 1 to 10

Exercise 1 – Read a horizontal shift

Describe how the graph of \(y=(x+3)^{2}\) is related to \(y=x^{2}\).

Exercise 2 – Read a vertical shift

Describe how \(y=\sqrt{x}-5\) is related to \(y=\sqrt{x}\).

Exercise 3 – Reflection

Describe how \(y=-|x|\) is related to \(y=|x|\).

Exercise 4 – Vertical stretch

Describe how \(y=2x^{3}\) is related to \(y=x^{3}\).

Exercise 5 – Move a point

The point \((2,5)\) is on \(f\). Find the matching point on \(y=f(x-4)+1\).

Exercise 6 – Reflect and shift a point

The point \((3,-2)\) is on \(f\). Find the matching point on \(y=-f(x+1)+3\).

Exercise 7 – Write an equation

Write the equation of the graph of \(y=x^{2}\) reflected over the \(x\)-axis, then shifted right 3 and down 2.

Exercise 8 – Vertex of a transformed parabola

Find the vertex of \(y=2(x+1)^{2}-6\) and tell whether it opens up or down.

Exercise 9 – Identify from a graph

Parent and transformed absolute value graphs.

The orange graph is a transformation of \(y=|x|\). Write its equation.

Exercise 10 – Horizontal changes of a point

The point \((4,6)\) is on \(f\) and \((3,-1)\) is on \(f\). Find the matching points on \(y=f(2x)\) and \(y=f\left(\frac{x}{3}\right)\) respectively.

Exercises 11 and up

Exercise 11 – Even or odd?

Decide whether each function is even, odd or neither: (a) \(f(x)=x^{4}-3x^{2}\); (b) \(g(x)=x^{3}+x\); (c) \(h(x)=x^{2}+x\).

Exercise 12 – Sequence of transformations

Describe the transformations that turn \(y=\sqrt{x}\) into \(y=-3\sqrt{x-2}+1\), and find the value at \(x=6\).

Exercise 13 – Domain and range after shifts

State the domain and range of \(y=\sqrt{x+4}-3\).

Exercise 14 – Transform a table

Values of f.

Make a table for \(g(x)=f(x-2)+1\).

Exercise 15 – Reflect over the y-axis

Give the domain of \(y=\sqrt{-x}\) and the matching point for \((9,3)\) on \(y=\sqrt{x}\).

Exercise 16 – Equation from a graph

A parabola from a transformation of y = x^2.

The parabola has vertex \((2,3)\), opens down and passes through \((3,2)\). Write its equation in the form \(y=a(x-h)^{2}+k\).

Exercise 17 – Fountain

Water path of Sofia’s fountain.

Water from Sofia’s garden fountain follows \(y=-0.5(x-4)^{2}+8\), with \(x\) and \(y\) in feet. Find the greatest height, the horizontal distance where it lands at \(y=0\) (besides \(x=0\)), and describe the transformations of \(y=x^{2}\).

Exercise 18 – Taxi fare transformation

A taxi fare is \(f(m)=2.5m\) dollars for \(m\) miles. A new company charges \(N(m)=0.8(2.5m)+4\). Describe the transformation of \(f\) and find \(N(10)\).

Exercise 19 – Who is right?

Two absolute value graphs.

Maya says \(y=|3x+2|\) compresses \(y=|x|\) horizontally by \(\frac13\) and then shifts it left 2. Is she right? Write the correct equation.

Exercise 20 – All features of a transformed parabola

For \(g(x)=-2(x+1)^{2}+5\), find the vertex, the range, the \(y\)-intercept and the \(x\)-intercepts to the nearest hundredth.

Key facts behind this precalculus transformations worksheet

Key facts

f(x) + k shifts up k; f(x – h) shifts right h.
-f(x) reflects over the x-axis; f(-x) reflects over the y-axis.
a f(x) stretches vertically by |a|; f(bx) compresses horizontally by 1/|b|.
A point (p, q) on f becomes (p + h, a q + k) on a f(x – h) + k.
Even functions satisfy f(-x) = f(x); odd functions satisfy f(-x) = -f(x).

How to use this precalculus transformations worksheet

Work through the exercises in order: they go from simple to challenging. Write every step on paper, then compare with the solutions page and redo any exercise you missed the next day.

Download the precalculus transformations worksheet as a PDF with the button below and keep it with your notes.

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