Vertical Dilation and Reflection: f(x) = ab^x
Grade 9 students in California Reveal Math Algebra 1 learn how the coefficient a in f(x)=ab^x controls vertical dilation and reflection of exponential functions. When |a|>1 the graph is vertically stretched, when 0<|a|<1 it is vertically compressed, and when a<0 the entire graph is reflected across the x-axis. The y-intercept always shifts to (0,a) and the horizontal asymptote stays at y=0. For example, f(x)=3·2^x stretches the parent y=2^x by a factor of 3 with y-intercept at (0,3), while f(x)=-2·3^x both stretches and reflects the parent function downward.
Key Concepts
For an exponential function of the form $f(x) = ab^x$, the value of $a$ determines a vertical dilation (stretch or compression) and, when $a < 0$, a reflection across the $x$ axis.
If $|a| 1$, the graph is vertically stretched by a factor of $|a|$. If $0 < |a| < 1$, the graph is vertically compressed by a factor of $|a|$. If $a < 0$, the graph is reflected across the $x$ axis .
Common Questions
What does the coefficient a do in f(x)=ab^x?
The coefficient a controls vertical dilation. If |a|>1, the graph is vertically stretched. If 0<|a|<1, the graph is vertically compressed. If a<0, the graph is also reflected across the x-axis.
Where is the y-intercept of f(x)=ab^x?
The y-intercept of f(x)=ab^x is always at the point (0,a). For example, f(x)=3·2^x has its y-intercept at (0,3) instead of the parent function's (0,1).
Does a vertical dilation change the horizontal asymptote?
No. The horizontal asymptote remains at y=0 regardless of the value of a. Only the y-intercept and the graph's width/orientation change.
What happens to an exponential growth function when a is negative?
When a is negative, the graph is reflected across the x-axis. An exponential growth function like y=3^x becomes a decreasing curve that approaches y=0 from below when multiplied by a negative a.
Can you give an example of a vertical compression?
f(x)=(1/4)·2^x is a vertical compression of the parent y=2^x by a factor of 1/4. The y-intercept shifts from (0,1) to (0,1/4) and the graph is wider.
Which unit covers this exponential transformation skill?
This skill is from Unit 8: Exponential Functions in California Reveal Math Algebra 1, Grade 9.