Exponential tables and graphs | Year 10

01 / LEARN

The idea, step by step

Multiply for each equal step

For $b > 0$ and $b \neq 1$ , the function $y = b^{x}$ multiplies $y$ by $b$ whenever $x$ increases by $1$ . For $b > 1$ the graph increases; for $0 < b < 1$ it decreases.

A zero exponent gives $y = 1$ when $x = 0$ . Negative inputs give positive reciprocals, so the graph remains above the horizontal axis.

Read the whole shape

The horizontal axis $y = 0$ is an asymptote of the basic function: values approach it without reaching it. The real domain is all real $x$ , and the range is $y > 0$ .

Plot enough points on both sides of zero and draw a smooth curve. Do not join the points as a staircase; the real-number function also has values between integers.

A common trap

The graph reaches $y = 0$ when $x$ is large and negative.

Instead: For a positive base, $b^{x} > 0$ for every real $x$ . Zero may be approached, not reached.

02 / WORKED EXAMPLES

See the thinking

Example 1

A growth table

For $y = 2^{x}$ , find the values at $x = - 2, - 1, 0, 1, 2, 3$ .

x	−2	−1	0	1	2	3
y	1/4	1/2	1	2	4	8

Use reciprocal powers for negative $x$ .
The outputs in order are $1 / 4, 1 / 2, 1, 2, 4, 8$ .
The graph passes through $(0, 1)$ and increases.

1 / 4, 1 / 2, 1, 2, 4, 8

Example 2

A decay intercept

Find the $y$ -intercept of $y = 5 {(1 / 2)}^{x}$ .

At the $y$ -axis, $x = 0$ .
$y = 5 {(1 / 2)}^{0} = 5$ .

(0, 5)

Check your understanding

What multiplier connects consecutive outputs of $y = 3^{x}$ ?

Show answer

$3^{x + 1} = 3 \times 3^{x}$ .

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Find $y$ when $x = - 1$ in $y = 4^{x}$ .
2 marks
Find the $y$ -intercept of $y = 3 \times 2^{x}$ .
2 marks
State whether $y = {0.7}^{x}$ increases or decreases as $x$ increases.
2 marks
For $y = 2^{x}$ , calculate the outputs at $x = 1 / 2$ and $x = 3 / 2$ exactly.
3 marks
State the domain and range of $y = 5^{x}$ .
2 marks
A culture starts with $4$ units and doubles each hour. Write its model and find its size after 3 hours.
3 marks

Read the worked solutions

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

Find $y$ when $x = - 1$ in $y = 4^{x}$ .
2 marks
$4^{- 1} = 1 / 4$ .
$1 / 4$
Find the $y$ -intercept of $y = 3 \times 2^{x}$ .
2 marks
Set $x = 0$ : $y = 3$ .
$(0, 3)$
State whether $y = {0.7}^{x}$ increases or decreases as $x$ increases.
2 marks
Each step multiplies by $0.7$ , a factor below $1$ .
Decreases
For $y = 2^{x}$ , calculate the outputs at $x = 1 / 2$ and $x = 3 / 2$ exactly.
3 marks
$2^{1 / 2} = \sqrt{2}$ and $2^{3 / 2} = 2 \sqrt{2}$ .
$\sqrt{2}$ and $2 \sqrt{2}$
State the domain and range of $y = 5^{x}$ .
2 marks
A positive-base exponential is defined for all real inputs and always positive.
Domain: all real $x$ ; range: $y > 0$ .
A culture starts with $4$ units and doubles each hour. Write its model and find its size after 3 hours.
3 marks
The initial multiplier is $4$ and the hourly factor is $2$ . At $t = 3$ , $N = 32$ .
$N = 4 \times 2^{t}$ ; 32 units.

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum Mathematics

Integer and fractional index laws; connecting fractional powers with roots; simplifying surds; operations with surds; rationalising denominators; exact and approximate values; exponential tables and graphs; growth/decay factors; comparing linear/quadratic/exponential patterns; exponential equations; logarithms as inverses; logarithm laws; interpreting logarithmic scales; graph transformations and parameter experiments; choosing and evaluating models.

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