Exponents and Logarithms

How to graph exponential and logarithmic functions?

Quick answer

To graph an exponential function y = b^x, plot the y-intercept (0, 1), sketch a curve approaching the horizontal asymptote y = 0, and rise if b > 1 or fall if 0 < b < 1. For a logarithmic function y = log_b(x), plot the x-intercept (1, 0), approach the vertical asymptote x = 0, and reflect across the diagonal line y = x.

The topic explained

How to graph exponential and logarithmic functions step by step

Exponential functions take the form y = b^x, where the base b is positive and not equal to 1. When b is greater than 1, like y = 2^x, the graph rises sharply from left to right, modeling exponential growth. When b is between 0 and 1, like y = (1/2)^x, the curve drops from left to right, modeling exponential decay. Every basic exponential function crosses the y-axis at (0, 1) because any non-zero number to the power of 0 equals 1. It also has a horizontal asymptote along the x-axis (y = 0) because b^x approaches zero but never touches it.

Logarithmic functions take the form y = log_b(x) and serve as the mathematical inverses of exponential functions. Because they reverse inputs and outputs, their graphs swap x and y coordinates. The base logarithmic graph passes through the x-intercept (1, 0) and features a vertical asymptote along the y-axis (x = 0). The domain is restricted to positive numbers (x > 0) because you cannot take the logarithm of zero or a negative value.

Graphing the two functions as inverses

Because y = b^x and y = log_b(x) are inverses, their graphs are exact reflections of each other across the line y = x. If you plot y = 2^x, you can mark points such as (-1, 0.5), (0, 1), (1, 2), and (2, 4). To plot y = log2(x), swap the coordinates to get (0.5, -1), (1, 0), (2, 1), and (4, 2). Drawing the diagonal dotted line y = x shows this symmetry clearly.

Watch out for the asymptotes when sketching by hand. Students often accidentally cross the horizontal asymptote y = 0 on an exponential curve or let a logarithmic curve cross the vertical line x = 0. Always draw the dashed asymptote first as a visual boundary before sketching your curve.

Free printable worksheet

Graphing Exponential and Logarithmic Functions

This worksheet is for 11th-grade math students. Identify the intercepts, asymptotes, and general shape for each curve before sketching.

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Graphing Exponential and Logarithmic FunctionsName:

State the y-intercept and horizontal asymptote for y = 3^x.
Does the graph of y = (1/3)^x rise or fall from left to right, and what is its horizontal asymptote?
Find the x-intercept, vertical asymptote, and domain for y = log3(x).
What is the relationship between the graph of y = 4^x and y = log4(x)?
Determine the vertical asymptote and domain for the transformed function y = log2(x - 3).
Determine the horizontal asymptote and y-intercept for y = 2^x - 4.

Common questions

Why does an exponential function have a horizontal asymptote instead of a vertical one?

An exponential function allows you to input any real number for x, so there are no vertical boundaries. As x heads toward negative infinity (for b > 1), b^x shrinks closer and closer to zero without ever reaching it, creating a horizontal boundary at y = 0.

How can you tell if a logarithmic graph is increasing or decreasing?

Look at the base. If base b is greater than 1, the logarithmic graph increases from left to right. If base b is between 0 and 1, the logarithmic graph decreases from left to right.

Can an exponential graph ever cross its horizontal asymptote?

The parent function y = b^x never crosses y = 0. While advanced rational functions can sometimes cross horizontal asymptotes, basic exponential functions approach the line continuously in one direction without ever touching or crossing it.

What changes when you add a constant inside the logarithm, like log(x + 2)?

Adding 2 inside the logarithm shifts the entire curve, including its vertical asymptote and domain, two units to the left. The vertical asymptote becomes x = -2 and the domain becomes x > -2.

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