Quadratic tables and graphs | Year 9

01 / LEARN

Understand the idea.

One input, one output

A table of values records the output for each chosen input. Substitute the entire signed number in brackets before evaluating powers: when $x = - 2$ , $x^{2} = {(- 2)}^{2} = 4$ . Then complete the remaining operations in the rule.

Each table row becomes an ordered pair $(x, y)$ . Plot input horizontally and output vertically. The table samples the relationship; for a quadratic with unrestricted real inputs, there are points between the listed ones as well.

See the shape, not a zigzag

A quadratic graph is a smooth parabola. Include inputs on both sides of its turning point to see the bend. Joining a few points with straight segments produces a misleading zigzag instead of the actual curve.

Label axes and choose scales that show the intercepts and turning point clearly. Symmetric outputs help check the table. If a point looks inconsistent, recalculate it before changing the graph to fit an error.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Worked example

Complete a table for $y = x^{2} - 1$ at $x = - 2, - 1, 0, 1, 2$ and describe the graph.

Coordinate graph of y=x squared minus one, x from minus four to four and y from minus four to ten. Horizontal grid intervals are one unit; vertical minor grid intervals are one unit.

Worked solution

1
Square every input.
$\begin{aligned} {(- 2)}^{2} & = 4, & {(- 1)}^{2} & = 1, \\ 0^{2} & = 0, & 1^{2} & = 1, & 2^{2} & = 4 \end{aligned}$
The negative inputs are squared as whole numbers.
2
Subtract one from each square.
$\begin{aligned} y (- 2) & = 3, & y (- 1) & = 0, \\ y (0) & = - 1, & y (1) & = 0, & y (2) & = 3 \end{aligned}$
The rule applies the same final subtraction to every input.
3
Plot the ordered pairs.
$(- 2, 3), (- 1, 0), (0, - 1), (1, 0), (2, 3)$
Each input pairs with its corresponding output.
4
Describe the complete curve.
$x^{2} \geq 0 \Rightarrow y = x^{2} - 1 \geq - 1$
The graph is a smooth upward-opening parabola with minimum (0,-1), consistent with its symmetric table.

Answer

$y = 3, 0, - 1, 0, 3$ ; a smooth upward parabola.

EXAMPLE 02

Worked example

Find the missing output for $y = x^{2} - 2 x$ when $x = - 3$ .

Worked solution

1
Replace every x with the signed input.
$y = {(- 3)}^{2} - 2 (- 3)$
Brackets preserve the negative sign in both occurrences of x.
2
Evaluate the power and product.
$y = 9 + 6 = 15$
Squaring a negative gives a positive, and subtracting a negative adds.
3
Write the point in coordinate order.
$(- 3, 15)$
The input is the first coordinate and the calculated output is the second.

Answer

$y = 15$

03 / YOUR TURN

Try it yourself.

Check your understanding

For $y = x^{2} + 2$ , find $y$ when $x = - 3$ .

Show answer & reasoning

$11$

Substitute $x = - 3$ using brackets: $y = {(- 3)}^{2} + 2 = 9 + 2 = 11$ . The point on the curve is $(- 3, 11)$ .

Keep practising

6 worksheet questions · 16 marks

Open worksheet Worked solutions

Read the questions online

For $y = x^{2} + 1$ , find $y$ at $x = - 2, 0, 2$ .
2 marks
For $y = x^{2} - 3 x$ , find $y$ when $x = - 2$ .
2 marks
Complete the outputs for $y = x^{2} - 4$ at $x = - 3, - 2, - 1, 0, 1, 2, 3$ .
3 marks
Use the supplied graph of $y = x^{2} - 4$ to read its two points on the $x$ -axis.
2 marks
Sketch $y = x^{2} + 1$ using at least five points. State the lowest point.
4 marks
Why is joining only $(- 2, 4)$ and $(2, 4)$ with a straight line not a sketch of $y = x^{2}$ ?
3 marks

Read the worked solutions

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

For $y = x^{2} + 1$ , find $y$ at $x = - 2, 0, 2$ .
2 marks
Square the inputs to get $4, 0, 4$ , then add $1$ .
$5, 1, 5$
For $y = x^{2} - 3 x$ , find $y$ when $x = - 2$ .
2 marks
${(- 2)}^{2} - 3 (- 2) = 4 + 6 = 10$ .
$10$
Complete the outputs for $y = x^{2} - 4$ at $x = - 3, - 2, - 1, 0, 1, 2, 3$ .
3 marks
Square each input, then subtract $4$ .
$5, 0, - 3, - 4, - 3, 0, 5$
Use the supplied graph of $y = x^{2} - 4$ to read its two points on the $x$ -axis.
2 marks
On the $x$ -axis, $y = 0$ ; the curve crosses at $x = - 2, 2$ .
$(- 2, 0)$ and $(2, 0)$
Sketch $y = x^{2} + 1$ using at least five points. State the lowest point.
4 marks
A useful table is $(- 2, 5), (- 1, 2), (0, 1), (1, 2), (2, 5)$ .
Join with a smooth upward curve.
Lowest point $(0, 1)$ .
Why is joining only $(- 2, 4)$ and $(2, 4)$ with a straight line not a sketch of $y = x^{2}$ ?
3 marks
At $x = 0$ the rule gives $0$ , while the straight join gives $4$ .
More points are needed to show the turn.
It misses the curved shape and the point $(0, 0)$ .

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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 ↗

In this lesson

Understand the idea.

One input, one output

See the shape, not a zigzag

See the method in action.

Worked example

Square every input.

Subtract one from each square.

Plot the ordered pairs.

Describe the complete curve.

Worked example

Replace every x with the signed input.

Evaluate the power and product.

Write the point in coordinate order.

Try it yourself.

Check your understanding

Keep practising