Parameter changes and transformations using graphing tools | Year 9

01 / LEARN

Understand the idea.

Shift the basic curve

In turning-point form, $y = {(x - h)}^{2} + k$ , the square reaches its smallest value at $x = h$ . The turn is therefore $(h, k)$ , and the symmetry axis is $x = h$ . This gives a reliable way to read horizontal shifts.

A positive $h$ shifts the basic parabola right; a negative $h$ shifts it left. The outside constant $k$ shifts every output vertically. For ${(x - 3)}^{2} - 2$ , the square vanishes at $3$ , giving minimum $- 2$ .

Change the coefficient

In $y = a {(x - h)}^{2} + k$ , require $a \neq 0$ for a parabola. The sign of $a$ sets the opening. Its magnitude scales vertical distances from the turn: $∣ a ∣ > 1$ gives a narrower appearance on unchanged axes.

When graphing, keep the window and axis scales fixed and vary one parameter at a time. Use the rule to predict the turn and a few matching outputs; this distinguishes a genuine transformation from a changed display scale.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Worked example

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

Graph of y=(x+2) squared plus three, with x from minus four to six and y from minus six to twelve. Horizontal grid intervals are one unit; vertical intervals are two units.

Worked solution

1
Find where the bracket vanishes.
$x + 2 = 0 \Rightarrow x = - 2$
The squared term has its minimum when the bracket is zero.
2
Calculate the minimum output.
$y = {(- 2 + 2)}^{2} + 3 = 3$
At that input, only the outside constant remains.
3
Identify the translation.
$(0, 0) ⟼ (- 2, 3)$
The basic turn moves two units left and three units up.
4
Check the scale and opening.
$a = 1$
The squared-term coefficient stays 1, so the translated curve retains its width and upward opening.

Answer

Two units left and three up; turning point $(- 2, 3)$ .

EXAMPLE 02

Worked example

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

Worked solution

1
Relate the outputs algebraically.
$y_{new} = - 2 y_{old}, y_{old} = x^{2}$
Every original output is multiplied by -2 for the same input.
2
Compare representative values.
$\begin{aligned} x = 1 : & 1 ⟼ - 2 \\ x = 2 : & 4 ⟼ - 8 \end{aligned}$
The sign reverses and each vertical magnitude doubles.
3
Check the turning point.
$- 2 {(0)}^{2} = 0, - 2 x^{2} \leq 0$
The turn stays at (0,0) and becomes a maximum.
4
Describe the transformation.
$a = - 2$
The graph reflects across the x-axis and stretches vertically by factor 2, so it opens downwards.

Answer

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

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

$(5, 2)$

The square in ${(x - 5)}^{2} + 2$ is zero at $x = 5$ , giving $y = 2$ . Since the square cannot be negative, this is the minimum turning point $(5, 2)$ .

Keep practising

6 worksheet questions · 14 marks

Open worksheet 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.

Checking saved progress…

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 ↗

In this lesson

Understand the idea.

Shift the basic curve

Change the coefficient

See the method in action.

Worked example

Find where the bracket vanishes.

Calculate the minimum output.

Identify the translation.

Check the scale and opening.

Worked example

Relate the outputs algebraically.

Compare representative values.

Check the turning point.

Describe the transformation.

Try it yourself.

Check your understanding

Keep practising