2.11 Transformations of graphs — Practise with help

Work each one first, closed book. Before you write the equation, decide whether the change acts on the input or on the output. Then track one point to check the direction.

Warm-up: remember

Work out $f(2)$ for $f(x) = x^2 - 3$ .
Complete the square for $x^2 - 6x + 5$ .
Work out $(2x)^2$ when $x = 3$ .

Answers

$f(2) = 2^2 - 3 = 1$ .
$x^2 - 6x + 5 = (x - 3)^2 - 4$ .
$(2 \times 3)^2 = 6^2 = 36$ .

If question 2 was hard, revisit completing the square: half the coefficient of $x$ gives the number in the bracket.

Practise with help

1. The curve $y = x^2$ is translated $5$ units to the right. Write the new equation.

Hint

A shift right acts on the input $x$ , so replace $x$ with $x - 5$ .

Next step

Write $y = (x - 5)^2$ , then check the vertex by asking where $x - 5 = 0$ .

Full working

Replace $x$ with $x - 5$ :

y = (x - 5)^2.

The vertex of the original curve is $(0, 0)$ . The new vertex is where $x - 5 = 0$ , so it is $(5, 0)$ : the curve moved right by $5$ , as intended.

2. The curve $y = x^3$ is reflected in the $x$ -axis. Write the new equation.

Hint

A reflection in the $x$ -axis changes the sign of the output.

Next step

Replace $y$ with $-y$ , so $y = -x^3$ .

Full working

The output changes sign, so

y = -x^3.

The point $(2, 8)$ on the original curve becomes $(2, -8)$ on the new curve.

3. Given $f(x) = x^2 + 2x$ , write the equation of $y = f(x + 3)$ .

Hint

Replace every $x$ in $f(x)$ with $x + 3$ , including the term $2x$ .

Next step

$f(x + 3) = (x + 3)^2 + 2(x + 3)$ . Expand both brackets.

Full working

Substitute $x + 3$ for $x$ :

f(x + 3) = (x + 3)^2 + 2(x + 3).

Expand:

(x + 3)^2 + 2(x + 3) = x^2 + 6x + 9 + 2x + 6 = x^2 + 8x + 15.

The equation is $y = x^2 + 8x + 15$ .

4. Given $f(x) = x^2$ , write the equation of $y = f(3x)$ in expanded form and describe the transformation.

Hint

Replacing $x$ with $3x$ acts on the input, so it is a horizontal change.

Next step

$f(3x) = (3x)^2 = 9x^2$ . A factor of $3$ inside gives a horizontal stretch by $\frac{1}{3}$ .

Full working

Substitute $3x$ for $x$ :

f(3x) = (3x)^2 = 9x^2.

The input is multiplied by $3$ , so the curve is squeezed horizontally by a factor of $\frac{1}{3}$ . The vertex stays at the origin, but the curve is narrower than $y = x^2$ .

5. The curve $y = x^2$ is translated $1$ unit to the left and then $4$ units downwards. Write the equation in completed-square form and state the vertex.

Hint

A shift left by $1$ replaces $x$ with $x + 1$ ; a shift down by $4$ subtracts $4$ from the output.

Next step

Start with $(x + 1)^2$ , then subtract $4$ : $y = (x + 1)^2 - 4$ .

Full working

Replace $x$ with $x + 1$ :

y = (x + 1)^2.

Subtract $4$ from the output:

y = (x + 1)^2 - 4.

The completed-square form is already written, so the vertex is at $(-1, -4)$ : one unit left and four units down from the origin.

6. A curve passes through the point $(2, 7)$ . The curve is then transformed to $y = 3f(x) + 1$ . Find the image of the point.

Hint

The transformation acts on the output. The $x$ -coordinate stays the same, and the $y$ -coordinate is changed by the rule.

Next step

Replace $y$ with $3y + 1$ : the new output is $3 \times 7 + 1$ .

Full working

The input $x = 2$ is unchanged. The output becomes

3f(2) + 1 = 3 \times 7 + 1 = 22.

The image of $(2, 7)$ is $(2, 22)$ .