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Here we will learn about **exponential graphs**, including recognising, sketching, plotting and interpreting exponential graphs.

There are also exponential graphs worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

An** exponential graph** is a representation of an exponential function of the form

\[y=k^x\]

Where **exponent **and **base**.

The y-intercept of an exponential curve (at x = 0 ) is 1 since **anything raised to the power 0 is 1**.

The **asymptote **to the curve.

The curve gets very close to the horizontal asymptote but does not touch it. This is because y ≠ 0.

The graph of an exponential function can represent either **exponential growth** or **exponential decay**:

- When
k is**greater than**it is a1 **growth**curve. Asx increases, so doesy .

It can be used to represent population growth or compound interest.

E.g.

\[y=2^x\]

- When
k is**less than**it is a1 **decay**curve. Asx increases,y decreases.

It can be used to represent radioactive decay.

E.g.

\[y=(\frac{1}{2})^x\]

There are more complex exponential functions of the form:

\[y=ab^x\]

The graphs look similar to the ones above, they have an exponent

E.g.

\[y=4\times 3^x\]

In order to recognise an exponential graph:

**Identify linear or quadratic or any other functions****Identify the exponential function****Identify your final answer**

Get your free exponential graph worksheet of 20+ questions and answers. Includes reasoning and applied questions.

COMING SOONGet your free exponential graph worksheet of 20+ questions and answers. Includes reasoning and applied questions.

COMING SOONThis lesson is part of the larger topic, types of graphs. It may be useful to first take a look at the main topic page, types of graphs, before exploring its related lessons in more detail:

Which is the correct equation for the graph?

\[y=x^2+3 \quad \quad \quad y=3x+1 \quad \quad \quad y=\frac{3}{x} \quad \quad \quad y=3^x\]

**Identify linear or quadratic or any other functions**

y=x^2+3 is a quadratic function; its graph would be a parabola.

y=3x+1 is a linear function; its graph would be a straight line.

y=\frac{3}{x} is a reciprocal function; its graph would be a hyperbola.

2**Identify the exponential function**

y=3^x is an exponential function.

It has

So the graph would be a growth curve.

3**Identify your final answer**

The correct equation for the graph is y=3^x

Which is the correct equation for the graph?

\[y=1-3x\quad\quad\quad y=x^2-3 \quad\quad\quad y=(\frac{1}{3})^x \quad\quad\quad y=x^{3}
\]

**Identify linear or quadratic or any other functions**

y=1-3x is a linear function; its graph would be a straight line.

y=x^2-3 is a quadratic function; its graph would be a parabola.

y=x^3 is a cubic function.

**Identify the exponential function**

y=(\frac{1}{3})^x is an exponential function.

It has

So the graph would be a decay curve.

**Identify your final answer**

The correct equation for the graph is y=(\frac{1}{3})^x

In order to plot an exponential graph:

**Complete the table of values****Plot the coordinates****Draw a smooth curve through the points**

Draw the curve of the equation for -1\leq{x}\leq3

y=2^x**Complete the table of values**

\[\begin{aligned}
&x \quad \quad \quad -1 \quad \quad \quad 0 \quad \quad \quad 1 \quad \quad \quad 2 \quad \quad \quad 3 \\\\
&y \quad \quad \quad \;\:0.5 \quad \quad \quad 1 \quad \quad \quad 2 \quad \quad \quad 4 \quad \quad \quad 8
\end{aligned}\]

**Plot the coordinates**

Draw an

**Draw a smooth curve through the points**

Draw the curve of the equation for -2\leq{x}\leq2

y=(\frac{1}{2})^x**Complete the table of values**

\[\begin{aligned}
&x \quad \quad \quad -2 \quad \quad \quad -1 \quad \quad \quad 0 \quad \quad \quad 1 \quad \quad \quad 2 \\\\
&y \quad \quad \quad \quad 4 \quad \quad \quad \quad \; 2 \quad \quad \quad \;1 \quad \quad \;\; 0.5 \quad \quad 0.25
\end{aligned}\]

**Plot the coordinates**

**Draw a smooth curve through the points**

In order to find the equation of an exponential graph:

**Substitute the pairs of values into the given equation****Solve the two simultaneous equations****Write down the equation of the exponential function**

The sketch shows a curve with equation y = ab^x where

Find the equation of the curve.

**Substitute the pairs of values into the given equation**

The given equation is:

\[y=ab^x\]

Substituting

\[2=ab^0\]

Substituting

\[162=ab^4\]

**Solve the two simultaneous equations**

\[2=ab^0\]

Since anything to the power

Now we look at the second equation:

\[162=ab^4\]

But we know that

\[162=2\times b^4\]

This can be solved to find the base

\[\begin{align*}
162&= 2\times b^4 \\\\
81 &= b^4 \\\\
\sqrt[4]{81} &= b \\\\
b & = 3
\end{align*}\]

**Write down the equation of the exponential function**

The original equation was:

\[y=ab^x\]

We found that

\[y=2\times 3^x\]

The sketch shows a curve with equation y = ab^x where

Find the equation of the curve.

**Substitute the pairs of values into the given equation**

The given equation is:

\[y=ab^x\]

Substituting

\[10=ab^1\]

Substituting

\[40=ab^3\]

**Solve the two simultaneous equations**

\[10=ab^1\]

Since anything to the power

\[10 = ab\]

Now we look at the second equation:

\[40=ab^3\]

We can solve these two equations by dividing one by the other:

\[\frac{40}{10}=\frac{ab^3}{ab}\]

This becomes:

\[4=b^2\]

This can be solved to find the base

\[\begin{align*}
4&= b^2 \\\\
\sqrt{4} &= b \\\\
b & = 2
\end{align*}
\]

We can substitute b = 2 into one of the equations to find a:

\[\begin{align*}
10 &= ab \\\\
10 &= a\times 2\\\\
a & = 5
\end{align*}\]

**Write down the equation of the exponential function**

The original equation was:

\[y=ab^x\]

We found that

\[y=5\times 2^x\]

**Curves should be smooth**

Do NOT join up the plotted points with straight lines like this:

The points should be joined by a single, smooth curve.

**All points should be on the curve**

If there is a point that does not join up with the other points when a smooth curve is drawn, this point is incorrect.

1. Identify the correct equation for the graph:

y=6^x

y=6x+1

y=x^2 +1

y=\frac{6}{x}

The curve is a growth curve; its equation will be an exponential function.

y=6^x is an exponential function.

y=6x+1 is a linear function; its graph would be a straight line.

y=x^2+1 is a quadratic function; its graph would be a parabola.

y=\frac{6}{x} is a reciprocal function; its graph would be a hyperbola.

2. Identify the correct equation for the graph:

y=1-10x

y=0.1^x

y=1-x^2

y=x^3+1

The curve is a decay curve; its equation will be an exponential function.

y=0.1^x is an exponential function.

y=1-10x is a linear function; its graph would be a straight line.

y=x^2+1 is a quadratic function; its graph would be a parabola.

y=x^3+1 is a cubic function.

3. Identify the correct equation for the graph:

y=10^x

A correct table of values would be:

\begin{aligned} &x \quad \quad 0 \quad \quad 1 \quad \quad \;\;2 \\ &y \quad \quad 1 \quad \quad 10 \quad \quad 100 \end{aligned}

These x values and y values give coordinates such as (0, 1), (1, 10) and (2,100).

The coordinates are plotted on the grid.

A smooth curve should be drawn through the points.

4. Identify the correct graph for the equation:

y=(\frac{1}{5})^x

A correct table of values would be:

\begin{aligned} &x \quad \quad -2 \quad \quad -1 \quad \quad 0 \quad \quad 1 \\ &y \quad \quad \;\;25 \;\;\;\quad \quad 5 \quad \quad \;1 \quad \;\;\; 0.2 \end{aligned}

These x values and y values give coordinates such as (-2, 25), (-1, 5) and so on.

The coordinates are plotted on the grid.

A smooth curve should be drawn through the points.

5. Find the equation of the curve in the form

y=ab^x

where a and b are constants and b > 0.

y=3\times 4^x

y=4\times 3^x

y=2\times 4^x

y=4\times 2^x

Substituting the coordinates into the given equation gives

4=ab^0=a

Now we need to find the base b

\begin{aligned} 32 &= ab^3 \\\\ 32 &= 4\times b^3 \\\\ 8 &= b^3\\\\ b &= \sqrt[3]{8}\\\\ b &= 2 \end{aligned}

So the equation is:

y=4\times 2^x

6. Find the equation of the curve in the form

y=ab^x

where a and b are constants and b > 0.

y=7\times 3^x

y=4\times 3^x

y=2\times 4^x

y=3\times 4^x

Substituting the coordinates into the given equation gives

21=ab^1=ab

189=ab^3

Dividing the two equations

\begin{aligned} \frac{189}{21} &= \frac{ab^3}{ab} \\\\ 9 &= b^2 \\\\ b &= \sqrt{9}\\\\ b &= 3 \end{aligned}

We can substitute b = 3 into one of the equations to find a.

\begin{aligned} 21 &= ab \\ 21 &= a\times 7\\ a & = 7 \end{aligned}

So the equation is:

y=7\times 3^x

1. Match the correct graph to the equation.

y=8^{x}

**(1 Mark)**

Show answer

Graph C

**(1)**

2. (a) Complete the table of values for y=4^x

\begin{aligned} &x \quad \quad -1 \quad \quad 0 \quad \quad 1 \quad \quad 2 \\ &y \end{aligned}

(b) On the grid, draw the graph of y=4^{x} for -1\leq{x}\leq2

**(4 Marks)**

Show answer

(a)

Correct values are: 0.25, 1, 4 and 16

For 1 correct y -value

**(1)**

For all correct y -values

**(1)**

(b)

for plotting the 4 points correctly

**(1)**

for the smooth curve

**(1)**

3. Here is a sketch of part of the graph y=pq^{x} where q\lt0

The points (0,3), (2,k) and (4,1875) are all on the graph y=pq^{x}

Find the value of k.

**(4 Marks)**

Show answer

3=pq^0 = p .

for p = 3

**(1)**

1875=pq^4 = 3\times q^4 .

**(1)**

q=\sqrt[4]{625}=5 .

for q=5

**(1)**

k=3\times 5^2=75 .

for the correct value of k

**(1)**

You have now learned how to:

- Recognise graphs of exponential functions
- Draw graphs of exponential functions
- Interpret graphs of exponential functions

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