Sketch graphs of exponential and logarithmic functions, identify their key features, and recognise them as reflections of each other

What this dot point is asking

SEAB wants you to recognise and sketch the graphs of exponential functions $y = a^x$ and logarithmic functions $y = \log_a x$, to mark their key features (intercepts and asymptotes), and to understand that the two are inverses, so each graph is the reflection of the other in the line $y = x$. Knowing the shapes lets you interpret growth and decay and check answers from the equation-solving topics.

The answer

The exponential graph y equals a to the x

For a base $a > 1$, the graph of $y = a^x$:

• passes through $(0,\ 1)$, because $a^0 = 1$,
• is always positive and increasing, rising more and more steeply as $x$ grows,
• has the $x$-axis as a horizontal asymptote ($y = 0$): the curve gets ever closer to it for large negative $x$ but never touches it.

If $0 < a < 1$ the curve instead decreases (models decay), still passing through $(0, 1)$ with the same asymptote $y = 0$.

The logarithmic graph y equals log of x

For base $a > 1$, the graph of $y = \log_a x$:

• passes through $(1,\ 0)$, because $\log_a 1 = 0$,
• is increasing but flattens out, growing more and more slowly,
• has the $y$-axis as a vertical asymptote ($x = 0$): the curve plunges downward as $x$ approaches $0$ from the right,
• is defined only for $x > 0$, because you cannot take the log of zero or a negative number.

The reflection relationship

Because $y = \log_a x$ is the inverse of $y = a^x$, the two graphs are mirror images in the line $y = x$. This swaps their features neatly: the exponential's $y$-intercept $(0, 1)$ becomes the log's $x$-intercept $(1, 0)$, and the exponential's horizontal asymptote $y = 0$ becomes the log's vertical asymptote $x = 0$.

Transformations

Adding a constant shifts the curve: $y = a^x + k$ moves the exponential up by $k$, which also moves its asymptote to $y = k$. Recognising a shift helps you read graphs in modelling questions.

Examples in context

Example 1. Modelling growth and decay. An increasing exponential models a growing investment or population; a decreasing exponential ($0 < a < 1$) models radioactive decay or cooling. The shape of the curve, read off the graph, tells you at a glance whether a quantity is growing or shrinking and how fast.

Example 2. Checking an equation solution. When you solve $2^x = 20$ and get $x \approx 4.32$, a quick sketch of $y = 2^x$ confirms the answer is sensible: the curve passes $16$ at $x = 4$ and $32$ at $x = 5$, so a value just above $4$ is right. The graph is a fast sanity check on the algebra.

Try this

Q1. State the coordinates where $y = 5^x$ crosses the $y$-axis. [1 mark]

• Cue. At $(0, 1)$, since $5^0 = 1$.

Q2. Write down the equation of the asymptote of $y = \log_2 x$ and the domain of the function. [2 marks]

• Cue. Vertical asymptote $x = 0$; domain $x > 0$.

Q3. Explain why the graphs of $y = 4^x$ and $y = \log_4 x$ are reflections of each other in $y = x$. [2 marks]

• Cue. $\log_4 x$ is the inverse of $4^x$, and the graph of an inverse is the reflection of the original in the line $y = x$.