Graphing Exponential Functions and Identifying Their Properties

Q1. Draw the graph of the following function.

Background

Topic: Graphing Exponential Functions

This question tests your understanding of how to graph exponential functions and recognize their key features, such as intercepts, growth/decay behavior, and specific points on the graph.

Key Terms and Formulas

• Exponential Function: A function of the form $f(x) = a^x$, where $a > 0$ and $a \neq 1$.

• Growth vs. Decay: If $a > 1$, the function shows exponential growth; if $0 < a < 1$, it shows exponential decay.

• Key Points: The y-intercept is always at $(0, 1)$ for $f(x) = a^x$. Other points can be found by substituting values for $x$.

Step-by-Step Guidance

1. Identify the base $a$ of the exponential function you are graphing. For example, if the function is $f(x) = 6^x$, then $a = 6$.

2. Calculate and plot several key points, such as $f(0)$, $f(1)$, $f(-1)$, and $f(2)$. For $f(x) = 6^x$:

   • $f(0) = 1$

   • $f(1) = 6$

   • $f(-1) = 1/6$

   • $f(2) = 36$

3. Draw the curve through these points, noting that the graph increases rapidly for positive $x$ (exponential growth) and approaches the x-axis (but never touches it) for negative $x$.

4. Label the y-intercept and other calculated points on your graph for clarity.

Try solving on your own before revealing the answer!

Final Answer:

The graph shown matches $f(x) = 6^x$, with points $(0,1)$, $(1,6)$, $(-1,1/6)$, and $(2,36)$ labeled. The curve demonstrates exponential growth as $x$ increases.