Graphing Solution Sets to Two-Variable Exponential Equations
Graphing Exponential Functions
Exponential functions are of the form \$y = a^x\$, where \$a\$ is a positive constant. The graph of an exponential function:
- Rises rapidly if \$a > 1\$.
- Falls rapidly if \$0 < a < 1\$.
The graph of an exponential function is not a straight line. It curvesand never touchesthe x-axis (asymptote).
Graphing Solution Sets
To graph the solution set of an equation like \$y = a^x\$:
- Create a table of values for \$x\$ and \$y\$.
- Plot the points on a coordinate plane.
- Connect the points with a smooth curve.
| \$x\$ | \$y = 2^x\$ |
|---|---|
| -2 | 0.25 |
| -1 | 0.5 |
| 0 | 1 |
| 1 | 2 |
| 2 | 4 |
Placeholder(graph): Graph of the exponential function y = 2^x, showing points from the table and the curve passing through them.
Placeholder(graph): Graph of the exponential function y = 3^x, showing the point (1, 3) highlighted.
Placeholder(graph): Graph of y = 2^x and y = 8, showing the intersection point at (3, 8).
NoteGraphical solutions are often approximations. For exact solutions, algebraic methods are preferred.
Solving Exponential Inequalities Graphically
To solve an inequality like \$a^x > b\$:
- Graph \$y = a^x\$ and \$y = b\$.
- Identify the region where the graph of \$y = a^x\$ is above the graph of \$y = b\$.
Placeholder(graph): Graph of y = 2^x and y = 8, highlighting the region where y = 2^x is above y = 8 for x > 3.
Self reviewGraph the solution set of \$y = 3^x\$ and identify the equation from the graph. Then, solve \$3^x = 9\$ graphically.
Theory of KnowledgeHow do graphs help us understand the behavior of exponential functions? Can all mathematical problems be solved graphically?
