College Algebra Test Review: Exponential and Logarithmic Functions

Q4. For the function $f(x) = 3^{x+2} - 1$, sketch the graph and give the asymptote, domain, and range. Describe the transformations in getting from $g(x) = 3^x$ to $f(x)$.

Background

Topic: Exponential Functions and Transformations

This question tests your understanding of how exponential functions are graphed, how to identify their asymptotes, domain, and range, and how to describe transformations such as shifts and reflections.

Key Terms and Formulas:

• Exponential Function: $f(x) = a^{x}$, where $a > 0$ and $a \neq 1$

• Transformation: Changes to the function such as shifts, stretches, or reflections.

• Horizontal Shift: $f(x) = a^{x+h}$ shifts the graph left if $h > 0$.

• Vertical Shift: $f(x) = a^{x} + k$ shifts the graph up if $k > 0$, down if $k < 0$.

• Asymptote: A line that the graph approaches but never touches.

• Domain: All possible $x$ values.

• Range: All possible $y$ values.

Step-by-Step Guidance

1. Start with the base function $g(x) = 3^x$. This is an exponential function with a horizontal asymptote at $y = 0$.

2. Apply the horizontal shift: $x + 2$ means the graph shifts left by 2 units.

3. Apply the vertical shift: $-1$ means the graph shifts down by 1 unit. The new horizontal asymptote is at $y = -1$.

4. Determine the domain and range: Exponential functions have domain $(-\infty, \infty)$ and range $( ext{asymptote}, \infty)$.

Try solving on your own before revealing the answer!

Final Answer:

Asymptote: $y = -1$

Domain: $(-\infty, \infty)$

Range: $(-1, \infty)$

Transformations: Horizontal shift left 2 units, vertical shift down 1 unit.

Q5. For the function $f(x) = e^{-x} + 3$, sketch the graph and give the asymptote, domain, and range. Describe the transformations in getting from $g(x) = e^x$ to $f(x)$.

Background

Topic: Exponential Functions and Transformations

This question tests your ability to analyze exponential functions with base $e$, including identifying asymptotes, domain, range, and describing transformations such as reflections and shifts.

Key Terms and Formulas:

• Exponential Function: $f(x) = e^x$

• Reflection: $f(x) = e^{-x}$ reflects the graph across the $y$-axis.

• Vertical Shift: $+3$ shifts the graph up by 3 units.

• Asymptote: The horizontal line the graph approaches.

• Domain: All real numbers.

• Range: All $y$ values above the asymptote.

Step-by-Step Guidance

1. Start with the base function $g(x) = e^x$.

2. Apply the reflection: $e^{-x}$ reflects the graph across the $y$-axis.

3. Apply the vertical shift: $+3$ shifts the graph up by 3 units. The new horizontal asymptote is at $y = 3$.

4. Determine the domain and range: Exponential functions have domain $(-\infty, \infty)$ and range $( ext{asymptote}, \infty)$.

Try solving on your own before revealing the answer!

Final Answer:

Asymptote: $y = 3$

Domain: $(-\infty, \infty)$

Range: $(3, \infty)$

Transformations: Reflection about $y$-axis, vertical shift up 3 units.