Sketching by hand and with graphing tools | Year 10

01 / LEARN

The idea, step by step

A sketch records the important structure

Start with the direction of opening and the axis of symmetry. Mark the vertex, intercepts and one extra pair of symmetric points if needed. Use a smooth curve and label coordinates rather than drawing an unlabelled U shape.

The scale need not be identical on both axes, but it must be consistent along each axis. A graph should show the features needed for the question.

Check the rule and the window

When using a graphing tool, enter brackets around the whole shifted square. Generate a short table first, then compare the plotted points with the screen.

A default window can hide a root or flatten the turning point. Change the horizontal and vertical ranges deliberately, and report approximate coordinates only to a precision the tool supports.

A common trap

Joining a few table values with straight-line segments and calling that the full parabola.

Instead: A quadratic is a smooth curve. Use the points as guides and show its symmetry.

02 / WORKED EXAMPLES

See the thinking

Example 1

Sketch with a useful checklist

Sketch $y = {(x - 1)}^{2} - 4$ .

The graph opens upward with vertex $(1, - 4)$ and axis $x = 1$ .
Roots satisfy ${(x - 1)}^{2} = 4$ , giving $x = - 1, 3$ .
The y-intercept is $(0, - 3)$ ; the symmetric point is $(2, - 3)$ .
Draw a smooth curve through these features.

Upward parabola through

(- 1, 0), (0, - 3), (1, - 4), (2, - 3), (3, 0)

Example 2

A reflected and stretched parabola

Sketch $y = - 2 {(x + 1)}^{2} + 8$ .

The graph opens downward and has vertex $(- 1, 8)$ .
Roots satisfy ${(x + 1)}^{2} = 4$ , so $x = - 3, 1$ .
The y-intercept is $(0, 6)$ , reflected in the axis by $(- 2, 6)$ .

Downward parabola with axis

x = - 1

, maximum

8

, roots

- 3, 1

Check your understanding

Which features must a sketch of $y = {(x - 2)}^{2} + 1$ show?

Show answer

Vertex $(2, 1)$ , axis $x = 2$ , y-intercept $(0, 5)$ , no x-intercepts.

The minimum value is positive, so the graph stays above the x-axis.

03 / WORKSHEET

Your turn

6 questions · 17 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Sketch $y = x^{2} - 4$ . Label the vertex and all intercepts.
3 marks
Sketch $y = - {(x - 2)}^{2} + 4$ . Label the vertex and all intercepts.
3 marks
Sketch $y = x^{2} + 2$ . Explain why there are no x-intercepts.
3 marks
Describe how to obtain $y = - x^{2} + 3$ from $y = x^{2}$ , then state the vertex.
2 marks
The table is intended for $y = x^{2} - 2 x - 3$ . Identify the incorrect entry and correct it.
3 marks
One y-value is incorrect.
x −1 0 1 2 3
y 0 −3 −4 −1 0
A tool uses $- 5 \leq x \leq 5$ to graph $y = x^{2} - 10 x + 16$ and shows only one root. Find both roots and suggest a better horizontal window.
3 marks

One y-value is incorrect.
x	−1	0	1	2	3
y	0	−3	−4	−1	0

Read the worked solutions

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

Sketch $y = x^{2} - 4$ . Label the vertex and all intercepts.
3 marks
$x^{2} = 4$ gives the roots; the y-intercept equals the vertex.
Vertex $(0, - 4)$ ; roots $- 2, 2$ ; upward opening.
Sketch $y = - {(x - 2)}^{2} + 4$ . Label the vertex and all intercepts.
3 marks
Set ${(x - 2)}^{2} = 4$ for the roots.
Vertex $(2, 4)$ ; roots $0, 4$ ; y-intercept $(0, 0)$ ; downward opening.
Sketch $y = x^{2} + 2$ . Explain why there are no x-intercepts.
3 marks
A real square cannot be negative, so $x^{2} + 2$ cannot equal zero.
Vertex $(0, 2)$ , upward opening; $x^{2} + 2 \geq 2$ .
Describe how to obtain $y = - x^{2} + 3$ from $y = x^{2}$ , then state the vertex.
2 marks
The negative sign reverses y-values; adding $3$ shifts them upward.
Reflect in the x-axis and move up 3; vertex $(0, 3)$ .
The table is intended for $y = x^{2} - 2 x - 3$ . Identify the incorrect entry and correct it.
3 marks
One y-value is incorrect.
x −1 0 1 2 3
y 0 −3 −4 −1 0
Substitute $x = 2$ : $4 - 4 - 3 = - 3$ .
The correct table is symmetric about $x = 1$ .
At $x = 2$ , $y = - 3$ , not $- 1$ .
A tool uses $- 5 \leq x \leq 5$ to graph $y = x^{2} - 10 x + 16$ and shows only one root. Find both roots and suggest a better horizontal window.
3 marks
$(x - 2) (x - 8) = 0$ .
The second root is outside the original horizontal window; the axis is $x = 5$ .
Roots $2, 8$ ; for example $0 \leq x \leq 10$ .

One y-value is incorrect.
x	−1	0	1	2	3
y	0	−3	−4	−1	0

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Curriculum reference & lesson scope

Australian Curriculum v9.0 and South Australian Curriculum

Year 10 quadratics, including the fuller school-course algebra requested for Ascent: non-monic factorisation, completing the square and the quadratic formula. These lessons consolidate Year 9 monic quadratics and extend the available solving methods.

Official curriculum source ↗

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