Lesson 16.1 · 25 min

Graphs of Quadratics

Tables of values for $y = a x^{2} + b x + c$ , the shape of a parabola, and what $a$ , $b$ and $c$ do to it.

In this lesson you will learn to…

Complete a table of values for a quadratic function
Plot the points and join them with a smooth curve
Recognise the parabola shape and its line of symmetry
Read the $y$ -intercept, turning point and roots from a graph

Why this matters

Parabolas describe the path of a thrown ball, the shape of a satellite dish and the arch of a bridge. Being able to sketch one from its equation tells you where it is highest or lowest and where it crosses zero.

Reading the graph

Feature	How to find it
$y$ -intercept	the value of $c$
roots	where the curve meets the $x$ -axis
turning point	the lowest (or highest) point
line of symmetry	the vertical line through the turning point

A parabola can cross the $x$ -axis twice, touch it once, or miss it completely — so a quadratic equation can have two, one or no solutions.

Common mistake. $-3^2$ in a table should be worked out as $(-3)^2 = 9$ . Put brackets round negative $x$ values when you square them.

Key rules and formulas

A quadratic
$y = a x^{2} + b x + c, a \neq = 0$
The highest power of $x$ is 2. Its graph is a smooth U-shaped curve called a parabola.
Which way up?
If $a > 0$ the parabola is a ∪ with a minimum point. If $a < 0$ it is a ∩ with a maximum point.
Where it crosses
The curve crosses the $y$ -axis at $(0, c)$ . Where it crosses the $x$ -axis are the roots — the solutions of $a x^{2} + b x + c = 0$ .
Table of values
Work out each term on its own row ( $x^{2}$ , then $- 3 x$ , then $+ 4$ ) and add the column. It keeps negatives under control.

Try it

Quadratic curve explorer

a (how steep, and which way up)1

c (where it crosses the y-axis)0

Equation: y = 1x² + 0x + 0
Turning point: (0, 0)
Roots: one root at x = 0

Set $a = 1$ , $b = - 3$ , $c = 4$ to see the curve from the worked example. Make $a$ negative to flip it upside down; change $c$ to slide it up and down.

Worked examples

Worked example 1

A table of values

Complete a table for

y = x^{2} - 3 x + 4

from

x = - 3

x = 3

Try it yourself first, then reveal the solution one step at a time.

Practice: graphs of quadratics

8 questions

Eight questions on tables of values and the shape of a parabola.

Question 1 of 8Complete the table
Complete the table for $y = x^{2} + 2 x - 3$ .
x −3 −2 −1 0 1 2
y Row 1, −3 Row 1, −2 Row 1, −1 Row 1, 0 Row 1, 1 Row 1, 2
Question 2 of 8Choose all that apply
From that table, select every true statement about $y = x^{2} + 2 x - 3$ .
Select every correct answer — there may be more than one.
It is an upside-down (∩) curveIt crosses the $x$ -axis at $x = - 3$ and $x = 1$ It crosses the $y$ -axis at $(0, - 3)$ Its minimum point is $(- 1, - 4)$
Question 3 of 8Explore and match
Set the sliders to show $y = x^{2} - 3 x + 4$ .
Move the controls until you match the question, then press Check.
a (how steep, and which way up)1
b0
c (where it crosses the y-axis)0
Equation
y = 1x² + 0x + 0
Turning point
(0, 0)
Roots
one root at x = 0
Question 4 of 8Multiple choice
What shape is the graph of $y = - 2 x^{2} + 5$ ?
Question 5 of 8Type the answer
Find $y$ when $x = - 2$ for $y = 2 x^{2} - x + 1$ .
Your answer
Question 6 of 8Fill the gaps
For $y = a x^{2} + b x + c$ :
If $a$ is Gap a the curve is ∪-shaped. The curve crosses the $y$ -axis at $y =$ Gap b. The points where it crosses the $x$ -axis are called the Gap c.
Question 7 of 8Type the answer
A parabola has $y = 2$ at both $x = 1$ and $x = 2$ . What is the equation of its line of symmetry? Give the value of $x$ .
Your answer
Question 8 of 8True or false
When plotting a quadratic, you should join the points with straight lines.

In this lesson you will learn to…

Why this matters

Reading the graph

Key rules and formulas

A quadratic

Which way up?

Where it crosses

Table of values

Quadratic curve explorer

Worked examples

Practice: graphs of quadratics