Sketching by hand and with graphing tools | Year 10

01 / LEARN

Understand the idea.

A sketch records the important structure

A quadratic sketch should communicate opening, vertex, symmetry and intercepts. Calculate and label these first; a pair of points at equal distances from the symmetry axis helps guide the curve.

Draw a smooth parabola through the features, using consistent spacing along each axis. The two axes may use different scales, but their labels must make the coordinates and the shape’s important features clear.

Check the rule and the window

Before trusting a graphing tool, check the entered rule against a few hand-calculated values. Brackets matter: squaring $x - 1$ requires the whole expression $(x - 1)$ inside the square.

Choose a window that contains the vertex and relevant intercepts. A cropped display can hide a root without removing it from the function. Use algebra for exact coordinates and label graphical readings as approximations.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Sketch with a useful checklist

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

Graph of y=(x-1) squared-4. The x window is -3 to 5 and the y window is -5 to 8. Fine grid lines are one unit apart; numbered labels are two units apart.

Worked solution

1
Locate the vertex and symmetry axis.
$\begin{array}{c} y = {(x - 1)}^{2} - 4 \\ vertex (1, - 4) \\ axis x = 1 \end{array}$
The positive square coefficient gives an upward-opening curve with a minimum.
2
Calculate the horizontal intercepts.
$\begin{aligned} {(x - 1)}^{2} & = 4 \\ x - 1 & = \pm 2 \\ x & = - 1 or 3 \end{aligned}$
The intercepts are $(- 1, 0)$ and $(3, 0)$ , equally spaced from the axis.
3
Add a symmetric pair near the vertex.
$y (0) = {(0 - 1)}^{2} - 4 = - 3, y (2) = {(2 - 1)}^{2} - 4 = - 3$
The inputs $0$ and $2$ are each one unit from $x = 1$ .
4
Sketch through the labelled points.
$\begin{array}{c} (- 1, 0), (0, - 3), (1, - 4) \\ (2, - 3), (3, 0) \end{array}$
A smooth upward curve through these points records both symmetry and the minimum.

Answer

Upward parabola through $(- 1, 0), (0, - 3), (1, - 4), (2, - 3), (3, 0)$ .

EXAMPLE 02

A reflected and stretched parabola

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

Graph of y=-2(x+1) squared+8. The x window is -4 to 2 and the y window is -4 to 10. Fine grid lines are one unit apart; numbered labels are two units apart.

Worked solution

1
Read the opening and vertex.
$\begin{array}{c} y = - 2 {(x + 1)}^{2} + 8 \\ vertex (- 1, 8) \\ axis x = - 1 \end{array}$
The coefficient is negative, so the curve opens downward and its maximum height is eight.
2
Solve for the roots.
$\begin{aligned} - 2 {(x + 1)}^{2} + 8 & = 0 \\ {(x + 1)}^{2} & = 4 \\ x & = - 3 or 1 \end{aligned}$
Both square-root branches are needed to locate the two intercepts.
3
Find a further symmetric pair.
$\begin{aligned} y (0) & = - 2 {(1)}^{2} + 8 = 6 \\ y (- 2) & = - 2 {(- 1)}^{2} + 8 = 6 \end{aligned}$
The vertical intercept $(0, 6)$ reflects across $x = - 1$ to $(- 2, 6)$ .
4
Draw a smooth downward curve.
$\begin{array}{c} (- 3, 0), (- 2, 6), (- 1, 8) \\ (0, 6), (1, 0) \end{array}$
The labelled features fix the curve’s maximum, crossings and symmetry.

Answer

Downward parabola with axis $x = - 1$ , maximum $8$ , roots $- 3, 1$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

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

The minimum is at $(2, 1)$ and the axis is $x = 2$ . At $x = 0$ , $y = 5$ . Since ${(x - 2)}^{2} \geq 0$ , the output is always at least $1$ , so there are no horizontal intercepts.

Keep practising

6 worksheet questions · 17 marks

Open worksheet 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 ↗

In this lesson

Understand the idea.

A sketch records the important structure

Check the rule and the window

See the method in action.

Sketch with a useful checklist

Locate the vertex and symmetry axis.

Calculate the horizontal intercepts.

Add a symmetric pair near the vertex.

Sketch through the labelled points.

A reflected and stretched parabola

Read the opening and vertex.

Solve for the roots.

Find a further symmetric pair.

Draw a smooth downward curve.

Try it yourself.

Check your understanding

Keep practising