Graphing Exponential Functions
Graph exponential functions in Grade 9 algebra by plotting key points for y=aˣ, identifying the horizontal asymptote, and recognizing exponential growth versus decay patterns.
Key Concepts
Property To graph an exponential function, make a table of ordered pairs for different x values and plot the points. The graph will always be a smooth curve that approaches, but never touches, the x axis. Explanation Graphing reveals the function's signature 'swoop' curve. Pick a few x values (include negatives and zero!), calculate the y values, and connect the dots. The graph will either shoot up for growth ($b 1$) or dive down toward zero for decay ($0<b<1$), always staying on one side of the x axis. Examples For $y = 2(3)^x$, points like $( 1, \frac{2}{3})$, $(0, 2)$, and $(1, 6)$ form a growth curve rising to the right. For $y = 10(\frac{1}{2})^x$, points like $( 1, 20)$, $(0, 10)$, and $(1, 5)$ form a decay curve falling to the right. The graph of $y = (4)^x$ is a reflection of $y=4^x$ across the x axis, showing a downward facing curve.
Common Questions
What does the graph of an exponential function look like?
An exponential function y = aˣ produces a curved graph that approaches but never touches the x-axis (horizontal asymptote). When a > 1 the graph rises steeply (growth); when 0 < a < 1 it decreases toward zero (decay).
What key points should you plot when graphing an exponential function?
Always plot the y-intercept at (0, 1) since any base raised to 0 equals 1. Then compute a few values like x = -2, -1, 1, 2 to reveal the curve's shape and direction.
What is the horizontal asymptote of an exponential function and why does it exist?
For y = aˣ, the horizontal asymptote is y = 0 (the x-axis). This exists because exponential values can get infinitely close to zero but mathematically never reach it — the function never actually crosses the x-axis.