AQA Level 2 Further Maths · Level 2 Certificate · Chapter 3: Algebra III · Lesson 7

Graphs of exponential functions

About 85 minutesCalculator allowed63 marks

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Quick questions to start.

A1
Needed today
Solve $3^{x} = \frac{1}{27}$ .
[2 marks]
Answer
A2
GCSE Higher
Work out the value of $8^{- \frac{2}{3}}$ .
[2 marks]
Answer
A3
Last lesson
$f (x) = 3 x + 10$ . Work out an expression for $f^{- 1} (x)$ .
[2 marks]
Answer
A4
From chapter 3
$f (x) = x^{2} + 3$ for $- 2 \leq x \leq 4$ . Work out the range of $f$ .
[2 marks]
Answer

Example, then your turn

Level 2

Your teacher works through each example with you. Then try the one beside it.

Example 1

Work out the values of $y = 2^{- x}$ for $x = - 3, - 2, - 1, 0, 1$ . Draw the graph on the grid.

Hint

2^{- x}

x = - 3

2^{3}

. At

x = 1

it is

2^{- 1} = \frac{1}{2}

. Join the points with a smooth curve, not straight lines.

B1
Work out the values of $y = 3^{x}$ for $x = - 2, - 1, 0, 1, 2$ , as fractions where needed. Draw the graph on the grid.
[3 marks]
Answer $y = \dots\dots, \dots\dots, \dots\dots, \dots\dots, \dots\dots$

Example 2

Sketch the graph of $y = 2^{x} + 3$ on the axes. Show where it crosses the $y$ -axis and the line it gets closer to.

Hint

y = 2^{x} + 3

y = 2^{x}

moved up

3

. At

x = 0

2^{0} = 1

. As

x

gets very negative,

2^{x}

gets closer to

0

B2
Sketch the graph of $y = 3^{- x} - 2$ on the axes. Give the coordinates of the point where it crosses the $y$ -axis and the equation of the line it gets closer to.
[3 marks]
Answer $(\dots, \dots)$ $y = \dots\dots$

Example 3

The graph of $y = 3^{x}$ is drawn on the grid.

(a) Use the graph to estimate the solution of $3^{x} = 5$ .

(b) By drawing a suitable straight line, estimate the solution of $3^{x} = 6 - x$ .

Hint (a) Read across from

y = 5

to the curve, then down. (b) Draw

y = 6 - x

; the solution is the

x

-coordinate where it meets the curve.

B3
The graph of $y = 2^{x}$ is drawn on the grid.
(a)
Use the graph to estimate the solution of $2^{x} = 5$ .
[1 mark]
Answer $x = \dots\dots\dots$
(b)
By drawing a suitable straight line on the grid, solve $2^{x} = 2 x + 1$ . Give both solutions.
[3 marks]
Answer $x = \dots\dots$ and $x = \dots\dots$
Total for B3: 4 marks

Example 4

The curve $y = a b^{x}$ passes through $(1, 10)$ and $(3, 250)$ . Work out $a$ and $b$ .

Hint Put each point into

y = a b^{x}

. Divide one equation by the other: the

a

cancels.

B4
The curve $y = a b^{x}$ passes through $A (- 1, 10)$ and $B (2, 0.08)$ . Work out $a$ and $b$ .
[4 marks]
Answer $a = \dots\dots$ $b = \dots\dots$

Practice

Level 2

C1
The sketch shows the graphs of $y = 3^{x}$ , $y = 3^{- x}$ , $y = 3^{x} - 2$ and $y = 2 \times 3^{x}$ . Match each equation to $P$ , $Q$ , $R$ or $S$ . Give the four letters in the order of the equations above.
[2 marks]
Answer $\dots \dots \dots \dots$
C2
The point $(- 2, \frac{1}{25})$ lies on the curve $y = a^{x}$ , where $a > 0$ . Work out $a$ .
[2 marks]
Answer $a = \dots\dots$

C3
The number of cells in a sample after $t$ hours is $N = 200 \times 3^{t}$ . Work out the first whole number of hours after which there are more than $100000$ cells.
[3 marks]
Answer $\dots\dots$ hours
C4
Work out the coordinates of the points where the curve $y = 2^{x} - 8$ crosses the $x$ -axis and the $y$ -axis.
[2 marks]
Answer $x$ -axis $(\dots, \dots)$ $y$ -axis $(\dots, \dots)$
C5
Sketch $y = 3^{x}$ and $y = x^{2}$ on the axes. Hence state the number of solutions of $3^{x} = x^{2}$ .
[2 marks]
Answer $\dots\dots$ solution(s)
C6
The curve $y = 9 \times 3^{- x}$ passes through the point $(p, 81)$ . Work out the value of $p$ .
[2 marks]
Answer $p = \dots\dots$

Exam-style questions

AQA exam style

D1
Here is a sketch of a curve. Circle its equation.
$y = 2^{x} y = 2^{- x} y = - 2^{x} y = 2^{x} - 1$
[1 mark]
Answer

D2
The curve $y = a b^{- x}$ passes through the points $(1, 1.5)$ and $(- 1, 24)$ , where $a$ and $b$ are positive constants.
(a)
Work out the values of $a$ and $b$ . You must show your working.
[4 marks]
Answer $a = \dots\dots$ $b = \dots\dots$
(b)
Work out the value of $y$ when $x = - 1.5$ .
[2 marks]
Answer $y = \dots\dots$
(c)
Hence show that the equation of the curve can be written as $y = 6 \times 0.2 5^{x}$ .
[2 marks]
Total for D2: 8 marks
D3
Josh says, "The curves $y = 3^{x}$ and $y = x^{3}$ only meet at $x = 3$ ."
Josh is wrong.
(a)
Work out $3^{x} - x^{3}$ when $x = 2$ and when $x = 2.5$ . Use your answers to show that Josh is wrong.
[2 marks]
(b)
Work out the $x$ -coordinate of this other point, correct to 1 decimal place. You must show your working.
[2 marks]
Answer $x = \dots\dots\dots$
Total for D3: 4 marks
D4
The number of fish in a lake $t$ years after the start of 2020 is modelled by $N = 3000 \times 0. 8^{t}$ .
(a)
Work out the number of fish the model gives for the start of 2023.
[2 marks]
Answer $\dots\dots$ fish
(b)
In which year does the number of fish first fall below $1000$ ?
[3 marks]
Answer $\dots\dots$
(c)
The number of fish in a second lake is modelled by $M = 500 \times 1.2 5^{t}$ . Show that when $N = M$ , $1.562 5^{t} = 6$ .
[2 marks]
Total for D4: 7 marks
D5
The sketch shows the curve $y = 3^{x} + c$ , where $c$ is a constant. The curve passes through $(2, 4)$ . The dashed line $y = c$ is the line the curve gets closer to.
Not drawn accurately
(a)
Work out the value of $c$ .
[1 mark]
Answer $c = \dots\dots$
(b)
Work out the coordinates of the points where the curve crosses the axes. Give the $x$ -coordinate correct to 2 decimal places.
[3 marks]
Answer $(\dots, \dots)$ and $(\dots, \dots)$
Total for D5: 4 marks

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
Work out the coordinates of the points where the curve $y = 4^{x} - 10 \times 2^{x} + 16$ crosses the $x$ -axis.
[4 marks]
Hint 1 · What to try
$4^{x} = (2^{x})^{2}$ . Let $u = 2^{x}$ .
Hint 2 · The first line
The equation becomes $u^{2} - 10 u + 16 = 0$ .
Hint 3 · The full method
$(u - 2) (u - 8) = 0$ , so $2^{x} = 2$ or $2^{x} = 8$ , giving $x = 1$ or $x = 3$ .
Answer $(\dots, \dots)$ and $(\dots, \dots)$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can work out values of $a^{x}$ and $a^{- x}$ and draw the graph.
I can sketch graphs such as $y = 3^{x} + k$ and $y = 3^{- x}$ , showing where they cross the $y$ -axis.
I can find $a$ and $b$ in $y = a b^{x}$ from two points.
I can solve equations with $a^{x}$ from a graph and use exponential models.

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