Parameter changes and transformations using graphing tools | Year 9

01 / LEARN

The idea, step by step

Shift the basic curve

The rule $y = {(x - h)}^{2} + k$ has turning point $(h, k)$ . Subtracting $h$ inside the bracket moves the turn right to $x = h$ ; adding $k$ outside moves it up. Find the input that makes the squared bracket zero to check the horizontal shift.

For $y = {(x - 3)}^{2} - 2$ , setting $x = 3$ makes the squared term zero. The minimum is therefore $- 2$ at $(3, - 2)$ . The axis is $x = 3$ .

Change the coefficient

In $y = a {(x - h)}^{2} + k$ , a negative $a$ reflects the curve so it opens downwards. Larger $∣ a ∣$ produces greater vertical change for the same horizontal distance from the turn, giving a narrower-looking curve on unchanged axes.

Use equal axis scales when comparing widths. In a graphing tool, enter two rules together, change one parameter at a time, predict the new turn, and confirm with a small table. Record a rule that fits several examples rather than relying on one picture.

A common trap

$y = {(x - 4)}^{2}$ shifts left four units.

Instead: The squared bracket becomes zero at $x = 4$ , so the turning point moves right.

02 / WORKED EXAMPLES

See the thinking

Example 1

Worked example

Describe $y = {(x + 2)}^{2} + 3$ compared with $y = x^{2}$ .

The squared bracket is zero when $x = - 2$ .
The smallest square is $0$ , so the minimum output is $3$ .
The coefficient remains $1$ , so the opening and width are unchanged.

Two units left and three up; turning point

(- 2, 3)

Example 2

Worked example

Compare $y = - 2 x^{2}$ with $y = x^{2}$ .

At $x = 1$ , the outputs are $- 2$ and $1$ .
At $x = 2$ , they are $- 8$ and $4$ .
The turning point remains $(0, 0)$ .

It opens downwards and has double the vertical magnitude at each input.

Check your understanding

Give the turning point of $y = {(x - 5)}^{2} + 2$ .

Show answer

$(5, 2)$

The bracket is zero at $x = 5$ ; then $y = 2$ .

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

State the turning point and axis of $y = {(x - 1)}^{2} - 3$ .
2 marks
State the turning point of $y = {(x + 4)}^{2} - 2$ .
2 marks
Which opens downwards: $y = 3 x^{2}$ or $y = - x^{2}$ ?
2 marks
At $x = 2$ , compare the outputs of $y = x^{2}$ and $y = 3 x^{2}$ .
2 marks
Write a quadratic rule with the same shape as $y = x^{2}$ and turning point $(3, - 5)$ .
3 marks
A graphing experiment changes $y = x^{2} + k$ from $k = - 2$ to $k = 4$ . Describe what changes and what stays the same.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

State the turning point and axis of $y = {(x - 1)}^{2} - 3$ .
2 marks
The bracket is zero at $x = 1$ and the output there is $- 3$ .
$(1, - 3)$ ; $x = 1$ .
State the turning point of $y = {(x + 4)}^{2} - 2$ .
2 marks
$x + 4 = 0$ when $x = - 4$ .
$(- 4, - 2)$
Which opens downwards: $y = 3 x^{2}$ or $y = - x^{2}$ ?
2 marks
The coefficient of the squared term is negative.
$y = - x^{2}$
At $x = 2$ , compare the outputs of $y = x^{2}$ and $y = 3 x^{2}$ .
2 marks
$2^{2} = 4$ and $3 (2^{2}) = 12$ .
$4$ and $12$ ; the second is triple.
Write a quadratic rule with the same shape as $y = x^{2}$ and turning point $(3, - 5)$ .
3 marks
Use a bracket zero at $x = 3$ , then shift down $5$ .
$y = {(x - 3)}^{2} - 5$
A graphing experiment changes $y = x^{2} + k$ from $k = - 2$ to $k = 4$ . Describe what changes and what stays the same.
3 marks
The output increases by $4 - (- 2) = 6$ for every input.
Only the constant outside the square changes.
Every point moves up $6$ units; shape, axis $x = 0$ and upward opening stay the same.

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum Mathematics

AC9M9A02, AC9M9A04–A06: monic quadratic expressions, integer-root equations, graphs and models.

Official curriculum source ↗

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