Exponential tables and graphs | Year 10

01 / LEARN

Understand the idea.

Multiply for each equal step

For $y = b^{x}$ with $b > 0$ and $b \neq 1$ , increasing $x$ by $1$ multiplies the output by $b$ . This produces growth when $b > 1$ and decay when $0 < b < 1$ .

Build a table around $x = 0$ : the zero power gives $1$ , and negative powers give reciprocals. Moving backwards one input step divides by $b$ . The basic exponential therefore passes through $(0, 1)$ and always has positive outputs.

Read the whole shape

The basic real exponential has domain all real $x$ and range $y > 0$ . Its horizontal asymptote is $y = 0$ : the curve can approach that line arbitrarily closely but never meets it.

Plot table points and join them with a smooth curve when modelling real inputs. Include points on both sides of zero to reveal the shape. A table at integer inputs samples the curve; it does not make the function a staircase.

Graphs of y equals two to the x and y equals one half to the x for x from minus two to three, with the original y range minus one to ten. Both pass through zero comma one and remain positive; the x and y axes are numbered.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

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

Worked solution

1
Evaluate the negative powers.
$\begin{aligned} 2^{- 2} & = \frac{1}{2^{2}} = \frac{1}{4} \\ 2^{- 1} & = \frac{1}{2} \end{aligned}$
Negative indices give reciprocal powers.
2
Evaluate zero and positive powers.
$\begin{aligned} 2^{0} & = 1, & 2^{1} & = 2 \\ 2^{2} & = 4, & 2^{3} & = 8 \end{aligned}$
Each unit input step doubles the previous output.
3
List outputs in the requested input order.
$\frac{1}{4}, \frac{1}{2}, 1, 2, 4, 8$
These correspond respectively to $x = - 2, - 1, 0, 1, 2, 3$ .

Answer

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

EXAMPLE 02

A decay intercept

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

Worked solution

1
Locate the vertical axis.
$x = 0$
Every point on the $y$ -axis has horizontal coordinate zero.
2
Evaluate the function there.
$y = 5 {(\frac{1}{2})}^{0} = 5 (1) = 5$
A nonzero base raised to zero is $1$ .
3
State the intercept as a point.
$(0, 5)$
An intercept includes both the input coordinate and output coordinate.

Answer

$(0, 5)$

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

The factor is $3$ , since $3^{x + 1} = 3 \times 3^{x}$ . This compares outputs for an input increase of exactly $1$ .

Keep practising

6 worksheet questions · 14 marks

Open worksheet 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 $x = \frac{1}{2}$ and $x = \frac{3}{2}$ 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
$4^{- 1} = \frac{1}{4}$ .
$\frac{1}{4}$
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 $x = \frac{1}{2}$ and $x = \frac{3}{2}$ exactly.
3 marks
$2^{\frac{1}{2}} = \sqrt{2}$ and $2^{\frac{3}{2}} = 2 \sqrt{2}$ .
$\sqrt{2}$ 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.

Official curriculum source ↗

In this lesson

Understand the idea.

Multiply for each equal step

Read the whole shape

See the method in action.

A growth table

Evaluate the negative powers.

Evaluate zero and positive powers.

List outputs in the requested input order.

A decay intercept

Locate the vertical axis.

Evaluate the function there.

State the intercept as a point.

Try it yourself.

Check your understanding

Keep practising