Question 5
Which statement best describes a feature of the graph of the exponential function $y=e^x,\ x\in R$?
Not quite. As $x\to\infty$, $e^x$ grows without bound, so $e^x\to\infty$. The value $e$ is just the height of the graph at $x=1$. In 2025, 24.3% of Queensland students chose (A).
Not quite. $e^0=1$, so the graph passes through $(0,1)$. The graph reaches $y=e$ at $x=1$. In 2025, 3.9% of Queensland students chose (B).
Not quite. The asymptote is horizontal, $y=0$, because $e^x\to0$ as $x\to-\infty$. The graph crosses $x=0$ at $(0,1)$, so $x=0$ can’t be an asymptote. In 2025, 27.2% of Queensland students chose (C).
That’s it. Step through the hints below to see why. In 2025, 44.1% of Queensland students chose (D).
At $x=0$, $y=e^0=1$, so the graph passes through $(0,1)$, not $(0,e)$. The gradient there is also 1.
Far to the left, the curve flattens onto the asymptote $y=0$ but never touches it. The height and the gradient shrink together, and they stay equal.
The height and the gradient are the same number. The tangent always meets the $x$-axis exactly 1 unit to the left, so its gradient is $\dfrac{\text{height}}{1}$.
Start with the features you can read off the graph.
$\displaystyle x=0:\ y=e^0=1\qquad$$\displaystyle x=1:\ y=e^1=e$
So the $y$-intercept is $(0,1)$. That rules out (B).
Look at each end of the graph.
$\displaystyle x\to\infty:\ e^x\to\infty\qquad$$\displaystyle x\to-\infty:\ e^x\to0$
The limit at $\infty$ isn’t $e$, which rules out (A). The graph flattens towards $y=0$, a horizontal asymptote, not $x=0$. That rules out (C).
The gradient of the graph is the derivative.
$$\frac{dy}{dx}=\frac{d}{dx}\left(e^x\right)=e^x=y$$
So at every point, the gradient equals the $y$-value. This is the property that makes $e$ special. Move the point in the figure and compare the two numbers.
$$\frac{d}{dx}\left(e^x\right)=e^x$$
The gradient at every point equals the value of the function there. The other statements are false: $e^x\to\infty$ as $x\to\infty$, the graph passes through $(0,1)$, and its asymptote is $y=0$.
How Queensland students answered in 2025
From the QCAA Mathematical Methods subject report for 2025. The percentages can add to less than 100 because some students didn’t answer.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording is from the 2025 QCAA Mathematical Methods external assessment, © State of Queensland (QCAA) 2025, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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