Pure 6.1 - Exponential functions and graphs | Edexcel A-Level Mathematics

Read exponential graphs from their equations: recognise growth and decay, use the natural base $e$ , and find the intercept, range and asymptote after a transformation. A few exact points will tell you more than a screenful of rounded calculator values.

The multiplicative meaning of $a^x$

Can a curve fall for ever and still stay above a horizontal line? Start with $y=3^x$ . Moving one step to the right multiplies the height by $3$ ; moving left divides it by $3$ .

$x$	$-2$	$-1$	$0$	$1$	$2$
$3^x$	$\frac19$	$\frac13$	$1$	$3$	$9$

You already know

For $a>0$ , $a^0=1$ , $a^{-n}=1/a^n$ , and $a^{1/n}$ is the positive root of $a$ of order $n$ . The variable in $a^x$ is the exponent; in $x^3$ it is the base.

Fractional and irrational exponents fill the gaps between the integer inputs. A positive base gives one positive real output at every real input.

Exponential function

The non-constant exponential function $y=a^x$ has a fixed base $a>0$ , $a\ne1$ , and a real variable exponent $x$ . For example, $3^x$ is exponential; $x^3$ is a power function. The allowed positive-base boundary $a=1$ gives the constant function $y=1$ , rather than growth or decay.

The index laws explain how to build a table without retyping every power.

Equal steps give equal factors

For $a>0$ and real $x$ and $h$ ,

a^{x+h}=a^x a^h

In particular, increasing $x$ by $1$ multiplies the output by $a$ .
Not in the formula booklet: learn it.

Try the same calculation with a base below $1$ .

Worked example 1: reciprocal-base points

Question. Find the exact points on $y=(\frac14)^x$ at $x=-1,0,1$ .

Plan. Substitute each input, using the reciprocal rule for the negative power.

\begin{aligned} x=-1 &: \quad y=(\tfrac14)^{-1}=4 && \text{take the reciprocal} \\ x=0 &: \quad y=(\tfrac14)^0=1 && \text{zero power} \\ x=1 &: \quad y=(\tfrac14)^1=\tfrac14 && \text{first power} \end{aligned}

Why this step? A negative exponent reverses multiplication; it does not make the output negative.

Answer. $(-1,4)$ , $(0,1)$ , and $(1,\frac14)$ .

Check. Each successive height is the previous height multiplied by $\frac14$ : $4,1,\frac14$ .

Change only the base to its reciprocal. The same inputs now produce growth rather than decay.

Worked example 2: reversing the base

Question. Find the exact points on $y=4^x$ at $x=-1,0,1$ .

Plan. Use the same three index rules as before.

\begin{aligned} 4^{-1}&=\tfrac14 && \text{reciprocal} \\ 4^0&=1 && \text{zero power} \\ 4^1&=4 && \text{first power} \end{aligned}

Why this step? Keeping the inputs fixed isolates the effect of changing the base.

Answer. $(-1,\frac14)$ , $(0,1)$ , and $(1,4)$ .

Check. These heights reverse the first table: $(\frac14)^x=4^{-x}$ , so the graphs reflect in the $y$ -axis.

The point at zero survives every change of positive base.

Watch out

Wrong: $5^{-1}=-5$ . Right: $5^{-1}=\frac15$ . A positive base raised to any real power stays positive. A negative base such as $-4$ cannot define a real-valued graph for every real input, since $(-4)^{1/2}$ is not real.

Now change the base once more and make the points exact.

Growth and decay shapes

A base of $2$ doubles each height; a base of $\frac12$ halves it. Neither operation can turn a positive height into zero.

[DIAGRAM: asset_slug: p1_6_1_know_and_use_the_function_ax_and_its_graph_where_a_is_positive__diagram_01; asset_name: Growth and decay exponential graphs; caption: Reciprocal bases reverse the direction; description: Two panels compare y equals 2 to the power x and y equals one half to the power x. Both pass through (0,1), remain above the x-axis and approach the horizontal asymptote y equals 0.]

Horizontal asymptote

A horizontal line $y=L$ is an asymptote when a curve's height approaches $L$ as the input goes indefinitely in one direction. For $y=a^x$ , with $a>0$ , $a\ne1$ , the line is $y=0$ . These particular curves never reach it.

Read the ends of the graph as well as its middle. These graph facts are not in the formula booklet: learn them.

Feature	$a>1$	$0<a<1$
Direction as $x$ increases	increasing	decreasing
End approaching $y=0$	$x\to-\infty$	$x\to\infty$
Domain	all real $x$	all real $x$
Range	$y>0$	$y>0$
$y$ -intercept	$(0,1)$	$(0,1)$

Worked example 3: describing decay

Question. Describe the direction, intercept, range and asymptote of $y=(\frac13)^x$ .

Plan. Compare the base with $1$ , then use positivity and the value at zero.

\begin{aligned} 0<\tfrac13<1 &\quad\Rightarrow\quad \text{decreasing} && \text{each step multiplies by }\tfrac13 \\ y(0)&=1 && \text{zero power} \\ (\tfrac13)^x&>0 && \text{positive base} \end{aligned}

Why this step? The base is positive but below $1$ ; “positive base” alone does not mean growth.

Answer. Decreasing; intercept $(0,1)$ ; range $y>0$ ; horizontal asymptote $y=0$ , approached to the right.

Check. At $x=1,2$ , the heights are $\frac13,\frac19$ : smaller but still positive.

Change only the base from $\frac13$ to $3$ . This reverses the direction, while the common features stay the same.

Worked example 4: describing growth

Question. Describe the same four features for $y=3^x$ .

Plan. Use $3>1$ for direction and $3^0=1$ for the intercept.

\begin{aligned} 3>1 &\quad\Rightarrow\quad \text{increasing} && \text{successive heights triple} \\ y(0)&=1 && \text{intercept} \\ 3^{-n}&=\frac1{3^n}\to0 && \text{left-hand behaviour} \end{aligned}

Why this step? Negative inputs explain the left-hand tail; the curve has no negative outputs.

Answer. Increasing; intercept $(0,1)$ ; range $y>0$ ; horizontal asymptote $y=0$ , approached to the left.

Check. The first part's table has heights $\frac19,\frac13,1,3,9$ , in increasing order.

A sketch should preserve these exact features even when its scale is approximate.

Watch out

A calculator display of $0$ for a very small exponential value is rounding or underflow. It is not an intercept. For example, $2^{-10}=\frac1{1024}>0$ .

Change the growth base to $4$ for your own description.

The natural exponential $e^x$

The base $e=2.71828\ldots$ lies between $2$ and $3$ , so $e^x$ has the growth shape you already know. Its exact points include $(-1,1/e)$ , $(0,1)$ , $(1,e)$ and $(2,e^2)$ .

Natural exponential

The natural exponential function is $y=e^x$ , where $e=2.71828\ldots$ . It has domain all real numbers, range $y>0$ , intercept $(0,1)$ , and horizontal asymptote $y=0$ . These graph facts are not in the formula booklet: learn them.

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Pure 6.1 - Exponential functions and graphs

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The multiplicative meaning of $a^x$

The natural exponential $e^x$