Graph Exponential, Logarithmic,
and Trig Functions
Master the key features of \( a^x \), \( \log_a x \), and \( \sin x \): intercepts, end behavior, period, midline, and amplitude.
Algebra 2
Three Function Families You Need to Know
Exponential
\( f(x) = a \cdot b^x \) where \( b \gt 0, b \neq 1 \). Models rapid growth or decay.
Logarithmic
\( f(x) = \log_b x \). The inverse of an exponential: "What exponent gives me \( x \)?"
Trigonometric
\( f(x) = a\sin(bx + c) + d \). Models periodic (repeating) behavior like waves.
Key Vocabulary
1
Intercept: where the graph crosses an axis. The y-intercept is the value when \( x = 0 \).
2
Asymptote: a line the graph approaches but never touches. Exponentials and logs each have one.
3
End behavior: what happens to \( f(x) \) as \( x \to \infty \) or \( x \to -\infty \).
4
Amplitude: half the distance between the max and min of a trig function.
5
Period: the horizontal length of one full cycle. Midline: the horizontal line at the average of max and min.
The Exponential Function: \( f(x) = b^x \)
The Logarithmic Function: \( f(x) = \log_b x \)
Exponential and Logarithmic: Inverse Pair
Anatomy of \( y = a\sin(bx) + d \)
Amplitude = \( |a| \)
Height from midline to peak
Period = \( \frac{2\pi}{|b|} \)
Length of one full cycle
Midline = \( y = d \)
Horizontal center of oscillation
Growth vs. Decay vs. Periodic
| Exponential | Logarithmic | Trigonometric | |
|---|---|---|---|
| Shape | J-curve (rapid rise/fall) | Slow climb | Wave (repeating) |
| Asymptote | Horizontal: \( y = 0 \) | Vertical: \( x = 0 \) | None |
| Domain | All real numbers | \( x \gt 0 \) | All real numbers |
| Range | \( y \gt 0 \) | All real numbers | \( [d - |a|, d + |a|] \) |
| Key Feature | End behavior | End behavior | Period, amplitude |
How to Graph Any of These Functions
1
Identify the family: exponential (\( b^x \)), logarithmic (\( \log_b x \)), or trigonometric (\( \sin, \cos \)).
2
Find intercepts: set \( x = 0 \) for y-intercept, set \( y = 0 \) for x-intercept.
3
Locate asymptotes: horizontal for exponentials, vertical for logarithms, none for trig.
4
Determine end behavior (exp/log) or amplitude, period, midline (trig).
5
Plot key points and sketch a smooth curve through them.
Example 1: Graph \( f(x) = 2^x \)
Example 2: Graph \( g(x) = \log_2 x \)
Example 3: Graph \( h(x) = 3\sin(2x) + 1 \)
Common Mistakes to Avoid
✕ Forgetting the Asymptote
Drawing the exponential curve touching or crossing the x-axis. It approaches \( y = 0 \) but never reaches it.
✕ Swapping Axes for Logs
Placing a horizontal asymptote on a log graph. Logs have a vertical asymptote at \( x = 0 \).
✕ Wrong Period Formula
Using \( \frac{\pi}{b} \) or \( 2\pi b \) instead of \( \frac{2\pi}{|b|} \) for the period of sine and cosine.
Transformation Cheat Sheet
| Parameter | Effect | Example |
|---|---|---|
| \( a \) (vertical stretch) | Stretches/compresses vertically; reflects if negative | \( 3 \cdot 2^x \): 3x taller |
| \( b \) (base or frequency) | Controls growth rate (exp) or period (trig) | \( \sin(2x) \): period \( = \pi \) |
| \( c \) (horizontal shift) | Shifts graph left/right | \( \log_2(x - 3) \): right 3 |
| \( d \) (vertical shift) | Shifts graph up/down; new midline for trig | \( \sin(x) + 2 \): up 2 |
Summary: Reading the Graph
Exponential
Look for: y-intercept, horizontal asymptote, end behavior (growth if \( b \gt 1 \), decay if \( 0 \lt b \lt 1 \)).
Logarithmic
Look for: x-intercept, vertical asymptote, domain (\( x \gt 0 \)), slow growth.
Trigonometric
Look for: amplitude \( |a| \), period \( \frac{2\pi}{|b|} \), midline \( y = d \), max/min values.
Question 1 of 5
What is the y-intercept of \( f(x) = 5 \cdot 3^x \)?
Substitute \( x = 0 \): \( f(0) = 5 \cdot 3^0 = 5 \cdot 1 = 5 \). The y-intercept is \( (0, 5) \).
Question 2 of 5
What type of asymptote does \( g(x) = \log_3 x \) have, and where is it?
All basic logarithmic functions \( \log_b x \) have a vertical asymptote at \( x = 0 \). The function is undefined for \( x \le 0 \).
Question 3 of 5
What is the period of \( y = \sin(4x) \)?
The period is calculated as \( \frac{2\pi}{|b|} = \frac{2\pi}{4} = \frac{\pi}{2} \). A larger \( b \) value compresses the wave horizontally.
Question 4 of 5
For \( h(x) = -2\cos(x) + 3 \), what are the amplitude and midline?
\( \text{Amplitude} = |a| = |-2| = 2 \). The negative sign means a reflection, not a negative amplitude. \( \text{Midline} = y = d = 3 \).
Question 5 of 5
As \( x \to \infty \), what happens to \( f(x) = \left(\frac{1}{2}\right)^x \)?
Since \( 0 \lt b = \frac{1}{2} \lt 1 \), this is exponential decay. As \( x \to \infty \), the function approaches the horizontal asymptote \( y = 0 \).