Graphs of Exponential Functions

• Exponential growth occurs when $k > 1$ (for example, $y = 3^x$). The graph increases at an increasing rate, rising very steeply in the first quadrant as $x$ gets larger.
• Exponential decay occurs when $0 < k < 1$ (for example, $y = 0.5^x$). The graph decreases as $x$ increases, starting high in the second quadrant and dropping down smoothly to level out.
• A useful property is symmetry: the graph of $y = 0.5^x$ is a perfect reflection of $y = 2^x$ in the y-axis.

Essential Features: Intercepts and Asymptotes

When the AQA exam asks you to "sketch" an exponential graph, you must clearly show two critical features. The first is the y-intercept, which is always located at the coordinate $(0, 1)$. This happens because any non-zero base $k$ raised to the power of 0 equals 1 ($k^0 = 1$).

The second critical feature is the horizontal asymptote. An asymptote is a straight line that a curve gets infinitely close to, but never meets. For $y = k^x$, the x-axis (the line $y = 0$) is the asymptote. Crucially, your sketched curve must not touch or cross the x-axis.

Worked Example: Sketching an Exponential Growth Graph

Sketch the graph of $y = 3^x$.

Step 1: Identify the shape of the curve.

• Because $k = 3$, which is greater than 1, this is an exponential growth curve. It will slope upwards from left to right.

Step 2: Mark the y-intercept.

• Plot a point at $(0, 1)$ and clearly label it on the y-axis.

Step 3: Draw the asymptote and curve.

• Draw a smooth freehand line starting on the far left, just above the negative x-axis.
• Bring the curve up through $(0, 1)$ and make it rise steeply into the top-right quadrant. Leave a visible, tiny gap between your curve and the x-axis on the left.

Worked Example: Sketching an Exponential Decay Graph

Sketch the graph of $y = 0.5^x$.

Step 1: Identify the shape of the curve.

• Because $k = 0.5$, which is between 0 and 1 ($0 < k < 1$), this is an exponential decay curve. It will slope downwards from left to right.

Step 2: Mark the y-intercept.

• Plot a point at $(0, 1)$ and clearly label it on the y-axis.

Step 3: Draw the asymptote and curve.

• Draw a smooth freehand line starting high in the top-left quadrant (the second quadrant).
• Bring the curve down through $(0, 1)$ and make it level out along the positive x-axis. Leave a visible, tiny gap between your curve and the x-axis on the right.

Worked Example: Finding the Base from a Coordinate

A graph of $y = k^x$ passes through the point $(3, 64)$. Find the value of $k$.

Step 1: Substitute the given $x$ and $y$ values into the equation.

• $64 = k^3$

Step 2: Solve for $k$ by finding the cube root.

• $k = \sqrt[3]{64}$
• $k = 4$
• The final equation is $y = 4^x$.

Higher Tier Extension: Transforming the Graph

If the equation includes a multiplier, such as $y = ab^x$, the y-intercept shifts from $(0, 1)$ to $(0, a)$. If a constant is added, like $y = k^x + c$, the entire graph shifts vertically. The horizontal asymptote moves to the line $y = c$, and the y-intercept becomes $(0, 1 + c)$.