Mathematics » Functions I » Exponential Functions

# Sketching Graphs of the Form y = abx + q

## Sketching graphs of the form $$y=a{b}^{x}+q$$

In order to sketch graphs of functions of the form, $$y=a{b}^{x}+q$$, we need to determine four characteristics:

1. sign of $$a$$

2. $$y$$-intercept

3. $$x$$-intercept

4. asymptote

The following video shows some examples of sketching exponential functions.

## Example

### Question

Sketch the graph of $$g(x)=3\times {2}^{x}+2$$. Mark the intercept and the asymptote.

### Examine the standard form of the equation

From the equation we see that $$a>1$$, therefore the graph curves upwards. $$q>0$$ therefore the graph is shifted vertically upwards by $$\text{2}$$ units.

### Calculate the intercepts

For the $$y$$-intercept, let $$x=0$$:

\begin{align*} y& = 3\times {2}^{x}+2 \\ & = 3\times {2}^{0}+2 \\ & = 3+2 \\ & = 5 \end{align*}

This gives the point $$(0;5)$$.

For the $$x$$-intercept, let $$y=0$$:

\begin{align*} y& = 3\times {2}^{x}+2 \\ 0& = 3\times {2}^{x}+2 \\ -2& = 3\times {2}^{x} \\ {2}^{x}& = -\cfrac{2}{3} \end{align*}

There is no real solution, therefore there is no $$x$$-intercept.

### Determine the asymptote

The horizontal asymptote is the line $$y=2$$.

### Plot the points and sketch the graph

Domain: $$\{x:x\in \mathbb{R}\}$$

Range: $$\{g(x):g(x)>2\}$$

Note that there is no axis of symmetry for exponential functions.

## Example

### Question

Sketch the graph of $$y=-2\times {3}^{x}+6$$

### Examine the standard form of the equation

From the equation we see that $$a<0$$ therefore the graph curves downwards. $$q>0$$ therefore the graph is shifted vertically upwards by $$\text{6}$$ units.

### Calculate the intercepts

For the $$y$$-intercept, let $$x=0$$:

\begin{align*} y& = -2\times {3}^{x}+6 \\ & = -2\times {3}^{0}+6 \\ & = 4 \end{align*}

This gives the point $$(0;4)$$.

For the $$x$$-intercept, let $$y=0$$:

\begin{align*} y& = -2\times {3}^{x}+6 \\ 0& = -2\times {3}^{x}+6 \\ -6& = -2\times {3}^{x} \\ {3}^{1}& = {3}^{x} \\ \therefore x& = 1 \end{align*}

This gives the point $$(1;0)$$.

### Determine the asymptote

The horizontal asymptote is the line $$y=6$$.

### Plot the points and sketch the graph

Domain: $$\{x:x\in \mathbb{R}\}$$

Range: $$\{g(x):g(x)<6\}$$

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