What shape is the graph of y = x²?

a U-shaped curve. It is a parabola with a minimum point.

Which equation has a graph made of two separate curves?

y = 1/x. A reciprocal graph never crosses either axis.

What shape is the graph of y = −x²?

an upside-down U. The negative turns the parabola over.

Graphs of curves · Cambridge IGCSE Mathematics (9

In plain words

A straight-line graph comes from an equation with just x. Put an x² in and it bends into a U; an x³ makes an S; 1/x makes two separate curves that never touch the axes. Recognising the shape from the equation is half the work.

Three things to know

y = x² (a quadratic) is a U-shaped parabola; with a negative x² it is an upside-down U. y = x³ (a cubic) is S-shaped. y = 1/x (a reciprocal) is two curves in opposite corners. y = 2ˣ (an exponential) climbs faster and faster.
A graph crosses the y-axis where x = 0, and crosses the x-axis where y = 0. Those x values are the roots of the equation.
To solve an equation using a graph, see where the curve meets a line: the x values of the crossing points are the solutions.

Worked example

For y = x² − 2x − 3, find where the graph crosses the axes.

On the y-axis x = 0, so y = −3.
On the x-axis y = 0: x² − 2x − 3 = 0, so (x − 3)(x + 1) = 0.
It crosses at (0, −3), (3, 0) and (−1, 0).

Tips and tricks

Plot curves with a smooth freehand line through the points. Never join them with a ruler.
When you square a negative number in a table of values, the result is positive: (−3)² = 9.

5 questions, about 2 minutes.

It lands in your notebook with its questions as flashcards.

Graphs of curves: 5 questions and answers

These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.

What shape is the graph of y = x²?
- a straight line
- a U-shaped curve (the answer)
- an S-shaped curve
- a circle
It is a parabola with a minimum point.
Where does the graph of y = x² − 4 cross the x-axis?
- at 4 and −4
- at 2 and −2 (the answer)
- at 0
- at 16
x² = 4.
Where does the graph of y = x² + 3x − 5 cross the y-axis?
- at 3
- at 5
- at −5 (the answer)
- at 0
Put x = 0.
Which equation has a graph made of two separate curves?
- y = x²
- y = 2x + 1
- y = 1/x (the answer)
- y = x³
A reciprocal graph never crosses either axis.
What shape is the graph of y = −x²?
- a U
- an upside-down U (the answer)
- a straight line
- an S
The negative turns the parabola over.

5 questions

Tap an answer and you’ll see straight away whether it’s right, and why.

Worksheet

3 questions, 6 marks. Write your answers on paper, then check them.

Graphs of curves

Cambridge IGCSE Mathematics (9–1) 0980 · 6 marks · papermunch.org

Name ______________________________ Date ______________

Complete: for y = x² − 4, when x = −3, y = ? and when x = 1, y = ?[2]
Show answerHide answer
5 and −3.
I got this one
Describe the shape of the graph of y = −x² + 2.[2]
Show answerHide answer
An upside-down U (a parabola with a maximum point), crossing the y-axis at 2.
I got this one
The graph of y = x² − 2x − 3 has a turning point where x = 1. Find its coordinates.[2]
Show answerHide answer
(1, −4). 1 − 2 − 3.
I got this one

Answers: Graphs of curves

1. 5 and −3.
2. An upside-down U (a parabola with a maximum point), crossing the y-axis at 2.
3. (1, −4). 1 − 2 − 3.