Exponential Functions and Their Graphs

1) From an expression to a function: f(x)=a^x

An exponential function uses a constant base a raised to a variable exponent x: f(x)=a^x, where a>0 and a≠1. Thinking of it as a function means you can input any real number x and get an output f(x), then interpret that output on a graph.

Key features (what you should be able to state quickly)

• Domain: all real numbers, (-∞,∞). You can plug in any real x.
• Range: positive real numbers, (0,∞). Outputs are never 0 or negative.
• y-intercept: f(0)=a^0=1, so the graph always passes through (0,1).
• x-intercepts: none, because a^x is never 0.
• Monotonic behavior: depends on a. If a>1, the function increases; if 0<a<1, it decreases.
• Horizontal asymptote: y=0. The graph gets arbitrarily close to 0 for one direction of x, but never touches it.

On a graph, the point (0,1) anchors the curve, and the base a controls whether the curve rises to the right (growth) or falls to the right (decay).

2) Growth vs decay: comparing a>1 and 0<a<1

Case A: a>1 (growth)

When a>1, increasing x multiplies the output by a each time x increases by 1. The graph rises to the right and approaches 0 to the left.

Example: f(x)=2^x

Sketch cues: plot (0,1), then (1,2), (2,4). For negative x, the points are between 0 and 1, like (-1,1/2). Draw a smooth curve increasing, flattening toward y=0 as x→-∞.

Continue in our app.

• Listen to the audio with the screen off.
• Earn a certificate upon completion.
• Over 5000 courses for you to explore!

Or continue reading below...

Download the app

y  |            • (3,8)  (growth rises right)        y=0 is asymptote (left side approaches it)  |        • (2,4)  |     • (1,2)  |  • (0,1)  +------------------------------ x

Case B: 0<a<1 (decay)

When 0<a<1, each step of x to the right multiplies the output by a number less than 1, so outputs shrink. The graph falls to the right and grows large to the left.

Example: g(x)=(1/2)^x

Sketch cues: plot (0,1), then (1,1/2), (2,1/4). For negative x, values exceed 1, like (-1,2). Draw a smooth curve decreasing, flattening toward y=0 as x→∞.

y  |  • (-2,4)  |     • (-1,2)  |        • (0,1)  |            • (1,1/2)  |                 • (2,1/4)  +------------------------------ x            (decay falls right)        y=0 is asymptote (right side approaches it)

A quick comparison you can use on any graph

• If the curve passes through (0,1) and rises as you move right, then a>1 (growth).
• If it passes through (0,1) and falls as you move right, then 0<a<1 (decay).
• In both cases, the curve stays above the x-axis and never crosses it.

3) Transformations: f(x)=k·a^(x-h)+c

Many real models and graphing tasks use a transformed exponential function:

f(x)=k·a^(x-h)+c

This form lets you control the shape and position of the basic curve a^x.

Step-by-step: how each parameter changes the graph

• Horizontal shift (h): a^(x-h) shifts the graph right by h units (if h>0) or left (if h<0). The “anchor” point moves from (0,1) to (h,1) before other transformations.
• Vertical stretch/compression and reflection (k): multiplying by k scales all y-values. If |k|>1, the graph stretches away from the asymptote; if 0<|k|<1, it compresses. If k<0, the graph reflects across the horizontal line y=c after shifting (equivalently, reflect across the x-axis first if c=0).
• Vertical shift (c): adding c moves the entire graph up/down. This also moves the horizontal asymptote from y=0 to y=c.

Asymptote and intercepts in the transformed form

• Horizontal asymptote: y=c.
• Domain: still all real numbers.
• Range: depends on k. If k>0, then f(x)>c; if k<0, then f(x)<c.
• Convenient point: since a^0=1, plugging x=h gives f(h)=k·1+c=k+c. So (h, k+c) is a fast point to plot.

Worked transformation example (plotting efficiently)

Example: f(x)=3·2^(x-1)-4

• Start with 2^x.
• (x-1) shifts right 1. The anchor point becomes (1,1).
• Multiply by 3: y-values triple, so the anchor becomes (1,3).
• Subtract 4: shift down 4, so the anchor becomes (1,-1).
• Asymptote: y=-4.

Make a small table around x=h=1 to sketch:

Plot these points and draw a smooth increasing curve approaching y=-4 on the left.

4) Interpreting parameters in context

When a situation is modeled by f(x)=k·a^(x-h)+c, each parameter has a practical meaning tied to the graph.

• a (growth/decay factor per 1 unit of x): for each increase of 1 in x, the distance from the asymptote is multiplied by a (when k>0). If a>1, the curve moves away from the asymptote as x increases; if 0<a<1, it moves toward the asymptote.
• k (initial scale relative to the asymptote): at x=h, the vertical distance from the asymptote is k, because f(h)-c=k. So k sets the starting “gap” from the long-run level.
• h (time/position shift): identifies the input value where the model’s “baseline exponential part” equals 1. Graphically, it locates the point (h, k+c).
• c (long-run level): the horizontal asymptote y=c. Many contexts interpret this as a baseline level, floor/ceiling, or equilibrium value the function approaches.

Context example (approaching a baseline): Suppose a device cools toward room temperature 20°C and is modeled (in simplified form) by T(t)=60·(0.8)^(t-0)+20. Here c=20 is the long-run temperature, k=60 is the initial difference above room temperature at t=0, and a=0.8 means the difference from room temperature is multiplied by 0.8 each time unit.

5) Practice reading graphs of exponential functions

A checklist for identifying the function’s behavior from its graph

• Find the horizontal asymptote: what y-value does the curve level off toward? That is c.
• Decide growth vs decay: as you move to the right, does the curve rise away from the asymptote (growth) or fall toward it (decay)?
• Estimate a convenient point: locate where the curve is one “unit” above the asymptote if possible, or use the point where the curve crosses the y-axis to estimate parameters.
• Describe long-run behavior: state what happens as x→∞ and as x→-∞ relative to the asymptote.

Reading values from a graph (estimation steps)

When you need f(x) from a graph:

1. Locate the given x on the horizontal axis.
2. Move vertically to the curve.
3. Move horizontally to the y-axis and read the approximate y-value.
4. Check reasonableness: is the value above/below the asymptote as expected? Is it increasing/decreasing in the correct direction?

Example prompts (use the checklist)

• Prompt A: A curve passes through (0,1), increases to the right, and gets close to y=0 on the left. Identify growth/decay and the asymptote. (Growth; asymptote y=0.)
• Prompt B: A curve levels off near y=5 and stays above 5. As x increases, it approaches 5. Identify the asymptote and whether it is decay toward the asymptote. (Asymptote y=5; decay toward 5.)
• Prompt C: A curve levels off near y=-2 and rises rapidly as x increases; it is always above -2. Describe long-run behavior. (As x→-∞, f(x)→-2; as x→∞, f(x)→∞.)

Common graph-reading pitfalls to avoid

• Confusing the x-axis with the asymptote: the asymptote is y=c, not always y=0.
• Assuming the curve crosses the asymptote: it approaches but does not reach it.
• Forgetting that exponential graphs are smooth and never turn back: they are monotonic (increasing or decreasing) for valid bases a>0, a≠1.