LemonmathSelf-study sheet

Grade 9 · Lesson 6.1

The function y = ax² (a ≠ 0)

Name: Date: Time spent: min

Goals

Recognise the function $y=ax^2$ ( $a\neq0$ ) and find its value at a given point.
Make a table of values and draw the graph (a parabola) of $y=ax^2$ .
State the symmetry of the graph and its position depending on the sign of $a$ .
Solve some real-life problems involving the function $y=ax^2$ .

Remember

The function $y=ax^2$ ( $a\neq0$ ) is defined for all $x\in\mathbb{R}$ ; $x$ and $-x$ give the same value of $y$ .
Its graph is a parabola with vertex $O$ and axis of symmetry the $y$ -axis.
$a>0$ : the graph is above the $x$ -axis and $O$ is lowest; $a<0$ : the graph is below the $x$ -axis and $O$ is highest.
$M(x_0, y_0)$ is on the graph $\Leftrightarrow y_0=ax_0^2$ ; use this to find $a$ or to find points.
To draw the graph: make a table of values with opposite pairs of $x$ and join with a smooth curve.

Exercises

Exercise 1

Consider the function $y=3x^2$ . Find the value of $y$ when $x=-2$ , $x=0$ , $x=\dfrac13$ and $x=\sqrt3$ .

Exercise 2

Make a table of values and draw the graph of $y=-\dfrac12x^2$ for $x\in\{-4, -2, 0, 2, 4\}$ . Describe the position of the graph.

Exercise 3

Which of the points $A(1, 2)$ , $B(-2, 8)$ , $C(3, 18)$ , $D(-1, -2)$ lie on the graph of $y=2x^2$ ?

Exercise 4

The graph of $y=ax^2$ passes through the point $M(-3, 6)$ .

a) Find $a$ .

b) Find the points on the graph whose $y$ -coordinate is $24$ .

Exercise 5

An object is dropped from a height of $125$ m. The distance it has fallen after $t$ seconds is $s=5t^2$ (m).

a) After $3$ seconds, how far above the ground is the object?

b) How long does it take to reach the ground?

Exercise 6

In grade 9 science (the lesson on kinetic and potential energy), the kinetic energy of an object of mass $m$ (kg) moving at speed $v$ (m/s) is $W=\dfrac12mv^2$ (J). Suppose a ball has mass $0.4$ kg; then $W=0.2v^2$ .

a) Find the kinetic energy of the ball when $v=10$ m/s.

b) The ball has kinetic energy $45$ J. What is its speed?

Exercise 7

Consider the function $y=3x^2$ . (Revise square-root expressions.)

a) Find the $x$ -coordinates of the points on the graph whose $y$ -coordinate is $6$ .

b) Find the value of $y$ when $x=1-\sqrt2$ .

Exercise 8

(2.0 marks) The graph $(P)$ of the function $y=ax^2$ passes through the point $A(2, 2)$ .

a) Find $a$ .

b) Draw $(P)$ .

c) Find the points on $(P)$ whose $y$ -coordinate is twice their $x$ -coordinate.

Exercise 9 Challenge

Suppose an entrance arch has the shape of a parabola. It is $8$ m wide at the base and its top is $6$ m above the ground. Choose axes with the origin $O$ at the top of the arch and the $y$ -axis pointing upwards; then the arch is part of the graph of $y=ax^2$ . A lorry $3$ m wide and $5$ m high drives through the middle of the arch. Can the lorry get through?

Mini test

15 minutes, without looking back at the lesson. Part I: circle one letter; Part II: write T or F for each statement; Part III: write the answer in the box.

Part I. Multiple choice

Question 1. Which of the following functions has the form

y=ax^2

with

a\neq0

y=3x

.B.

y=-\dfrac12x^2

.C.

y=x^2+2

.D.

y=0x^2

Question 2. Consider the function

y=-2x^2

. The value of

y

when

x=-3

18

.B.

-18

.C.

36

.D.

6

Question 3. The graph of

y=ax^2

passes through the point

(-2, 8)

. The value of

a

2

.B.

-2

.C.

4

.D.

-4

Question 4. Which of the following points does not lie on the graph of

y=-x^2

(1, -1)

.B.

(-2, -4)

.C.

(3, 9)

.D.

(0, 0)

Part II. True or false

Question 1. The function

y=\dfrac13x^2

has graph

(P)

. Decide whether each statement is true or false.

a) $O(0, 0)$ is the lowest point of $(P)$ .	T / F
b) $(P)$ passes through the point $(3, 3)$ .	T / F
c) $(P)$ passes through the point $(-3, -3)$ .	T / F
d) The points of $(P)$ with $y$ -coordinate $12$ have $x$ -coordinates $6$ and $-6$ .	T / F

Part III. Short answer

Question 1. An object falls from a height of

80

m, and the distance fallen after

t

seconds is

s=5t^2

(m). After how many seconds does it reach the ground?

Question 2. The graph of

y=ax^2

passes through the point

A(4, 4)

. Find

a

(give your answer as a decimal).

Score: ........ / 10

Hints (only when stuck)

Exercise 1.

What you need: substitute the value of $x$ into the formula, square first, then multiply by $3$ .

Steps:

Work out $x^2$ for each value.
Multiply the result by $3$ .

Exercise 2.

What you need: if $a<0$ , the parabola lies below the $x$ -axis and $O$ is its highest point.

Steps:

Work out $y$ at each value of $x$ .
Plot the points and join them with a smooth curve.

Exercise 3.

What you need: $M(x_0, y_0)$ lies on the graph exactly when $y_0=2x_0^2$ .

Steps:

Work out $2x_0^2$ for each point and compare it with $y_0$ .

Exercise 4.

What you need: substitute the coordinates into $y=ax^2$ ; the equation $x^2=k$ ( $k>0$ ) has two solutions $\pm\sqrt k$ .

Steps:

Solve $9a=6$ .
Solve $ax^2=24$ .

Exercise 5.

What you need: the height above the ground is $125-s$ ; the object reaches the ground when $s=125$ .

Steps:

Work out $s$ when $t=3$ .
Solve $5t^2=125$ with $t>0$ .

Exercise 6.

What you need: $W=0.2v^2$ is a function of the form $y=ax^2$ with $a=0.2$ ; the speed $v>0$ .

Steps:

Substitute $v=10$ .
Solve $0.2v^2=45$ .

Exercise 7.

What you need: $x^2=2\Leftrightarrow x=\pm\sqrt2$ ; the identity $(a-b)^2=a^2-2ab+b^2$ .

Steps:

Solve $3x^2=6$ .
Expand $(1-\sqrt2)^2$ , then multiply by $3$ .

Exercise 8.

What you need: points on a graph; a product equation $A\cdot B=0$ (lesson equations reducible to linear equations).

Steps:

Solve $4a=2$ .
Make a table of values with $x\in\{-4, -2, 0, 2, 4\}$ .
The required points have the form $(x, 2x)$ : solve $\dfrac12x^2=2x$ .

Exercise 9.

What you need: the feet of the arch are the points $(-4, -6)$ and $(4, -6)$ .

Steps:

Find $a$ from the point $(4, -6)$ .
The sides of the lorry are $1.5$ m from the centre line: find $y$ at $x=1.5$ .
The height of the arch there is $6+y$ ; compare it with $5$ .

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