A-Level Exponential functions Practice Questions

Write out the power as repeated multiplication

A power tells us how many times to multiply the base by itself. The base is 3 and it appears 4 times, so we write it out in full.

Multiply the first three factors

Work left to right. Multiplying the first three 3s together gives 27.

Multiply by the last factor

Multiply the running total by the final 3 to reach the answer.

State the value

All four factors have been combined, so this is the value of the power.

Find the y-intercept

Setting $x=0$ gives $y=1$, so the correct curve must pass through $(0,1)$.

Check the direction

Because the base 2 is greater than 1, the curve should rise from left to right and stay above the $x$-axis.

Identify the asymptote

The correct graph hugs the $x$-axis on the left, never dropping below it.

Recognise a special value

The only way an exponential equals 1 is when its power is 0, because $e^{0}=1$ and $e$ to any non-zero power is not 1.

Set the powers equal

So the power $2x$ must equal 0.

Solve

Dividing both sides by 2 gives $x=0$.

Collect the working so far

Before writing the answer, we gather the results of the previous lines to make sure nothing has been dropped. The key quantity we have found is shown above.

State the final answer clearly

We now write the answer on its own line. Setting it out clearly makes it easy to read back and easy for an examiner to mark.

State the solution

This is the only solution, since the exponential curve meets the line $y=1$ just once.

Find the gradient function

The derivative of $e^{x}$ is itself.

Consider the sign of the derivative

The exponential function is always positive; it can get very small but never reaches 0 or goes negative.

Link gradient sign to behaviour

A curve is increasing wherever its gradient is positive. Since the gradient is positive everywhere, the curve rises for all $x$.

Restate what has been shown

We pause to state clearly the result our working has established. This is the statement we now match against the options.

Note it holds for all valid x

The argument above did not rely on a special value of $x$, so the conclusion holds generally. This generality is important for a correct proof or reasoning answer.

Link to the graph's behaviour

The conclusion is consistent with the shape of the exponential graph, which gives a helpful visual check on the algebra.

Test with a numerical example

Trying a simple number in the result confirms it behaves as claimed. A worked example like this reassures us the reasoning is right.

Identify the matching option

Comparing our conclusion with the list of statements, exactly one option says precisely this. That is the answer.

Reject the first misconception

One distractor comes from forgetting to bring the constant $k$ down when differentiating. Our working shows this is wrong.

Conclude

So the curve never levels off or turns downwards; it always climbs from left to right.

Find the y-intercept

Setting $x=0$ gives $y=1$, so the curve passes through $(0,1)$.

Find the asymptote

As $x$ grows, $e^{-x}\to0$, so $y$ rises towards the asymptote $y=2$.

Check behaviour for negative x

For very negative $x$, $e^{-x}$ is huge, so $y$ dives downwards; the curve is increasing overall.

List the features to match

To choose the right graph we make a short checklist of the curve's key features: where it crosses the $y$-axis, whether it rises or falls, and the position of its asymptote.

Check the y-intercept feature

The correct graph must cross the $y$-axis at the height we calculated above. This rules out any option with the wrong starting point.

Check the direction feature

The correct graph must rise or fall in the same way as our curve. A curve going the wrong way indicates a reflection, so it can be rejected.

Check the asymptote feature

The correct graph must flatten out towards the same horizontal line we found. An option with a different asymptote has been shifted vertically and is wrong.

Reject the reflected graph

One distractor is the curve reflected in an axis, so it goes the wrong way. Because its direction disagrees with our analysis, we reject it.

Reject the vertically shifted graph

Another distractor has been moved up or down, giving the wrong asymptote and intercept. This does not match, so we reject it.

Reject the horizontally shifted graph

A further distractor is shifted left or right, so its key point sits at the wrong $x$-value. We reject it too.

Reject the non-exponential graph

One option is not even the right type of curve (for example a straight line), so it cannot be an exponential graph and is rejected.

Confirm only one graph fits

Exactly one graph has the correct intercept, direction and asymptote all together, which pins down the answer.

Match to the correct graph

We select the graph that matches every feature in our checklist.

Double-check with a sample point

As a final check we test one easy point on the chosen graph and confirm it agrees with our equation. It does, so the choice is secure.

Find the x-intercept

The curve crosses the $x$-axis where $e^{-x}=2$, i.e. $x=-\ln2$, just left of the origin.