2.11 Transformations of graphs — Practise

Work through these on your own. For every transformation, say whether it acts on the input or the output, write the new equation, and track one point to check the direction.

Both the questions and the answers below come from a set that is checked by computer every time the book is built.

The curve $y = f(x)$ is translated $3$ units to the right. Write the equation of the new curve when $f(x) = x^2$ .
The curve $y = x^2$ is translated $2$ units upwards. Write the new equation.
The curve $y = x^2$ is stretched vertically by a factor of $3$ . Write the new equation.
The curve $y = x^2$ is stretched horizontally by a factor of $\frac{1}{2}$ . Write the new equation in expanded form.
Given $f(x) = x^3$ , write the equation of $y = 2f(x - 1) + 3$ in expanded form.
Given $f(x) = \sqrt{x}$ , write the equation of $y = f(x + 4)$ and state its domain.
The point $(2, 8)$ is on the curve $y = x^3$ . Write the equation of the curve after a reflection in the $x$ -axis, and state the image of the point.
Given $f(x) = 2x + 1$ , write the equation of $y = f(-x)$ .
Given $f(x) = x^2 + 1$ , write $y = f(2x) + 1$ and $y = 2f(x) + 1$ in expanded form and explain why they differ.
The curve $y = x^2$ is transformed into $y = (3x)^2$ . Describe the transformation in words, and give the transformed equation in expanded form.
The curve $y = x^2$ is translated $2$ units to the right and then $5$ units downwards. Write the equation of the new curve in completed-square form and state its vertex.
Compare $y = x^3$ and $y = -x^3 + 4$ . State the two transformations and the effect on the point $(1, 1)$ .

Answers and concise solutions

1. The new equation

Answer: A translation to the right by $3$ replaces $x$ with $x - 3$ . The new equation is $y = (x - 3)^2$ . The vertex moves from $(0, 0)$ to $(3, 0)$ . Expanding gives $x^2 - 6x + 9$ , but the factorised form shows the transformation.

2. The new equation

Answer: A translation upwards by $2$ adds $2$ to the output: $y = x^2 + 2$ . The vertex moves from $(0, 0)$ to $(0, 2)$ .

3. The new equation

Answer: A vertical stretch by $3$ multiplies the output: $y = 3x^2$ . Every point moves three times as far from the $x$ -axis, and the vertex stays at $(0, 0)$ .

4. The new equation

Answer: A horizontal stretch by a factor of $\frac{1}{2}$ multiplies the input by $2$ : replace $x$ with $2x$ . Then $y = (2x)^2 = 4x^2$ . A horizontal stretch changes the input, so it affects the $x$ -coordinate in the opposite way to what the factor suggests.

5. The new equation

Answer: Apply the transformations in order. Replace $x$ with $x - 1$ : $(x - 1)^3$ . Multiply by $2$ : $2(x - 1)^3$ . Add $3$ : $2(x - 1)^3 + 3$ . Expanding, $(x - 1)^3 = x^3 - 3x^2 + 3x - 1$ , so the expanded form is $2x^3 - 6x^2 + 6x + 1$ .

6. The new equation and the domain

Answer: Replace $x$ with $x + 4$ : $y = \sqrt{x + 4}$ . The expression under the root must not be negative, so $x + 4 \geq 0$ and the domain is $x \geq -4$ . The graph is the original curve moved $4$ units to the left.

7. The new equation

Answer: A reflection in the $x$ -axis sends $y$ to $-y$ , so the new equation is $y = -x^3$ . Each point keeps its $x$ -coordinate and its $y$ -coordinate changes sign, so $(2, 8)$ maps to $(2, -8)$ .

8. The new equation

Answer: Replace $x$ with $-x$ : $f(-x) = 2(-x) + 1 = -2x + 1$ . The line is reflected in the $y$ -axis, so its gradient changes sign.

9. Both equations

Answer: For $f(2x)$ the input is doubled first: $f(2x) = (2x)^2 + 1 = 4x^2 + 1$ , so $f(2x) + 1 = 4x^2 + 2$ . For $2f(x)$ the output is doubled: $2f(x) = 2(x^2 + 1) = 2x^2 + 2$ , so $2f(x) + 1 = 2x^2 + 3$ . The two differ because $f(2x)$ changes the input and $2f(x)$ changes the output. A stretch and a translation do not commute.

10. The description and the equation

Answer: The input has been multiplied by $3$ , which is a horizontal stretch by a factor of $\frac{1}{3}$ . The equation simplifies: $y = (3x)^2 = 9x^2$ . The curve is squeezed towards the $y$ -axis, so it becomes narrower than $y = x^2$ .

11. The equation and the vertex

Answer: Translating $2$ right replaces $x$ with $x - 2$ , giving $(x - 2)^2$ . Translating $5$ down subtracts $5$ : $y = (x - 2)^2 - 5$ . The completed-square form is already written, so the vertex is $(2, -5)$ .

12. The transformations and the image

Answer: Two transformations are applied to $y = x^3$ : a reflection in the $x$ -axis, giving $y = -x^3$ , then a translation $4$ units upwards, giving $y = -x^3 + 4$ . The point $(1, 1)$ reflects to $(1, -1)$ , then moves up to $(1, 3)$ .