Graphing Exponential Functions: Table of Values and Smooth Curve
Graphing exponential functions using a table of values is a Grade 9 Algebra 1 skill in California Reveal Math (Unit 8). Students build a five-point table including at least two negative x-values, x = 0, and two positive x-values, then connect them with a smooth curve that approaches but never crosses the horizontal asymptote y = 0. For f(x) = 3(1/2)^x, the table gives (−2, 12), (−1, 6), (0, 3), (1, 1.5), (2, 0.75) — the curve falls right and hugs the asymptote as x increases.
Key Concepts
To graph $f(x) = ab^x$, build a table using at least two negative $x$ values, $x = 0$, and at least two positive $x$ values. Plot each point $(x, f(x))$, then draw a smooth, continuous curve that:.
passes through the y intercept $(0,\, a)$ rises steeply (growth) or falls steeply (decay) on one side approaches but never crosses the horizontal asymptote $y = 0$ on the other side.
Common Questions
Why should you include negative x-values when graphing an exponential?
Negative x-values show the steep side of the curve where the function grows rapidly. Using only non-negative x-values misses the full exponential shape and asymptotic behavior.
What does the table look like for f(x) = 2^x?
f(-2) = 1/4, f(-1) = 1/2, f(0) = 1, f(1) = 2, f(2) = 4. Plot these five points and connect with a smooth curve that rises steeply to the right and approaches y = 0 to the left.
Why does the graph approach but never cross y = 0?
Because b^x is always positive for positive base b. The curve gets arbitrarily close to zero but can never equal zero or go negative. This boundary is the horizontal asymptote.
How do you read the y-intercept directly from the exponential formula f(x) = ab^x?
Substitute x = 0: f(0) = a*b^0 = a*1 = a. The y-intercept is always (0, a), which you can read directly from the leading coefficient a without calculation.
What common mistake makes exponential graphs look wrong?
Connecting points with straight line segments instead of a smooth curve. Exponential growth/decay is curved — straight segments produce an angular shape that fails to show the asymptotic behavior correctly.