Math Problem Statement

Complete the table of values for y = 2^x and identify which of the sketches represents y = 2^x.

Solution

Let's break down and solve the problem:

Part (a): Complete the table of values for y=2xy = 2^x

We need to calculate y=2xy = 2^x for each given xx-value.

  • When x=−1x = -1, y=2−1=12=0.5y = 2^{-1} = \frac{1}{2} = 0.5
  • When x=0x = 0, y=20=1y = 2^0 = 1
  • When x=1x = 1, y=21=2y = 2^1 = 2
  • When x=2x = 2, y=22=4y = 2^2 = 4
  • When x=3x = 3, y=23=8y = 2^3 = 8
  • When x=4x = 4, y=24=16y = 2^4 = 16
  • When x=5x = 5, y=25=32y = 2^5 = 32 (this value is already given)

Thus, the completed table is:

\hline x & -1 & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline y & 0.5 & 1 & 2 & 4 & 8 & 16 & 32 \\ \hline \end{array}$$ ### Part (b): Identify the graph of $$y = 2^x$$ This is an exponential function. Key characteristics of exponential graphs include: - The graph is always increasing for positive values of $$x$$. - As $$x \to -\infty$$, $$y \to 0$$, but it never touches the $$x$$-axis. - As $$x \to \infty$$, $$y \to \infty$$, and the graph steeply rises. Looking at the provided graphs: - **Graph A**: This shows an increasing curve, starting near the $$x$$-axis and rising sharply, which is typical of exponential growth. It appears to match $$y = 2^x$$. - **Graph B**: This graph appears to decrease for negative values of $$x$$, which is incorrect for $$y = 2^x$$. - **Graph C**: This graph decreases and does not represent an increasing exponential function. Thus, **Graph A** represents $$y = 2^x$$. --- Do you want further details or have any questions? Here are 5 related questions to expand your understanding: 1. How does changing the base in an exponential function affect its graph? 2. What would happen to the table if the function was $$y = 3^x$$ instead of $$y = 2^x$$? 3. How would you graph the function $$y = 2^{-x}$$? 4. What are the main characteristics of exponential decay functions? 5. How would the table change if you were asked to find values for negative powers of 2? **Tip:** Always look at the behavior of the graph as $$x \to -\infty$$ and $$x \to \infty$$ to understand the nature of exponential functions.

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Math Problem Analysis

Mathematical Concepts

Exponential Functions
Graphing
Table of Values

Formulas

Exponential formula y = 2^x

Theorems

Properties of Exponential Growth

Suitable Grade Level

Grades 8-10

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