Answers to Tutorials in Exponential Functions

This page provides the answers to the interactive tutorials found in Exponential Functions .

Interactive Tutorial – Part 1

2 – The function \( f(x) \) increases.

3 – The function \( f(x) \) increases.

4 – The function \( f(x) \) decreases.

5 – The function \( f(x) \) decreases.

Interactive Tutorial – Part 4

1. When \( |a| > 1 \) and increases, the graph is vertically stretched. When \( 0 < |a| < 1 \), the graph is vertically compressed. If \( a \) is negative, the graph is also reflected across the \( x \)-axis.
2. When \( |b| > 1 \), the graph is horizontally compressed. When \( 0 < |b| < 1 \), the graph is horizontally stretched. If \( b \) is negative, the graph is reflected across the \( y \)-axis.
3. When \( c \) is positive, the graph is shifted to the left. When \( c \) is negative, the graph is shifted to the right.
4. When \( d \) is positive, the graph is shifted upward. When \( d \) is negative, the graph is shifted downward.
5. The \( y \)-intercept is given by the point \( (0, f(0)) \). \[ f(0) = a B^{\,b(0 + c)} + d \] \[ f(0) = a B^{\,bc} + d \] The parameters that affect the \( y \)-intercept are \( a \), \( B \), \( b \), \( c \), and \( d \).

   Yes, the graph of an exponential function will always have a \( y \)-intercept.

6. To find the \( x \)-intercept, we solve \( f(x) = 0 \): \[ a B^{\,b(x + c)} + d = 0 \] Rearranging, \[ B^{\,b(x + c)} = -\frac{d}{a} \] This equation has solutions only if \( -\dfrac{d}{a} > 0 \). Taking logarithms, \[ b(x + c) = \log_{B}\!\left(-\frac{d}{a}\right) \] Solving for \( x \): \[ x = \frac{\log_{B}\!\left(-\frac{d}{a}\right)}{b} - c \] The parameters that affect the \( x \)-intercept (when it exists) are \( a \), \( B \), \( b \), \( c \), and \( d \).

   No, the graph will have an \( x \)-intercept only when \( -\dfrac{d}{a} \) is positive.