Graphing Exponential Functions: f(x) = 5^x with Coordinates

Math Problem Statement

Graph the given function by making a table of coordinates.

f left parenthesis x right parenthesis equals 5 Superscript xf(x)=5x

Question content area bottom

Part 1

Complete the table of coordinates.

minus−2

minus−1

StartFraction 1 Over 25 EndFraction125

one fifth15

2525

(Type integers or fractions. Simplify your answers.)

Part 2

Choose the correct graph below.

-1123456-2-112xy

A coordinate system has a horizontal x-axis labeled from negative 1 to 6 in increments of 1 and a vertical y-axis labeled from negative negative 2 to 2 in increments of 1. From left to right, a curve rises steeply in quadrant four passing through the plotted points (0.2, negative 1) and (0.4, negative 0.5), crosses the positive x-axis at plotted point (1, 0), and then rises at a decreasing rate in quadrant 1 passing through the plotted points (2.2, 0.5) and (5, 1). As x decreases towards 0, the curve approaches the negative y-axis from the right. All coordinates are approximate.

-3-2-112351015202530xy

A coordinate system has a horizontal x-axis labeled from negative 3 to 3 in increments of 1 and a vertical y-axis labeled from negative 1 to 30 in increments of 1. From left to right, a curve rises shallowly in quadrant 2 passing through the plotted points (negative 2, 0.2) and (negative 1, 0.4), crosses the positive y-axis at plotted point (0, 1), and then rises steeply in quadrant 1 passing through the plotted points (1, 2.2) and (2, 5). As x decreases towards negative infinity, the curve approaches the negative x-axis from above. All coordinates are approximate.

510152025-2-112xy

A coordinate system has a horizontal x-axis labeled from negative 1 to 26 in increments of 1 and a vertical y-axis labeled from negative negative 2 to 2 in increments of 1. From left to right, a curve rises steeply in quadrant four passing through the plotted points (0, negative 2.4), (0.2, negative 1.4), and (1, negative 0.4), crosses the x-axis at plotted point (2, 0), and then rises at a decreasing rate in quadrant 1 passing through a plotted point (25, 1.6). As x decreases towards 0, the curve approaches the negative y-axis from the right. All coordinates are approximate.

-3-2-112351015202530xy

Solution

To complete the table and choose the correct graph, let's analyze the function $f(x) = 5^x$ .

Part 1: Table of Coordinates

We'll calculate $f(x)$ for each value of $x$ :

For $x = -2$ : $f(-2) = 5^{-2} = \frac{1}{5^2} = \frac{1}{25}$
For $x = -1$ : $f(-1) = 5^{-1} = \frac{1}{5}$
For $x = 0$ : $f(0) = 5^0 = 1$
For $x = 1$ : $f(1) = 5^1 = 5$
For $x = 2$ : $f(2) = 5^2 = 25$

Completed Table:

\hline x & f(x) \\ \hline -2 & \frac{1}{25} \\ -1 & \frac{1}{5} \\ 0 & 1 \\ 1 & 5 \\ 2 & 25 \\ \hline \end{array}$$ --- ### Part 2: Graph Selection Key features of the graph: 1. **Exponential Growth**: The function grows rapidly as $$x$$ increases. 2. **Behavior as $$x \to -\infty$$**: The function approaches 0 but never reaches it (asymptote at $$y = 0$$). 3. **Key Points**: The graph should pass through the points: - $$(-2, \frac{1}{25})$$ - $$(-1, \frac{1}{5})$$ - $$(0, 1)$$ - $$(1, 5)$$ - $$(2, 25)$$ #### Comparing the Options: - **Option A**: Does not match because the curve appears to decline on the left incorrectly. - **Option B**: Matches the behavior of $$5^x$$. The curve passes through $$(-2, 0.04)$$, $$(-1, 0.2)$$, $$(0, 1)$$, $$(1, 5)$$, $$(2, 25)$$. - **Option C**: Incorrect. The exponential growth is steep but doesn't follow the correct pattern for $$5^x$$. - **Option D**: Same issue as C; does not fit. #### Correct Answer: **Option B** Let me know if you'd like a detailed graph or further clarification! --- ### 5 Related Questions: 1. What is the general form of an exponential function, and how does it behave? 2. How can you determine the asymptotes of an exponential graph? 3. What happens to the graph if the base of the exponential function changes? 4. How do you find the domain and range of an exponential function? 5. How can you use transformations to shift or reflect an exponential graph? ### Tip: When sketching exponential graphs, focus on key points and the horizontal asymptote to guide the shape accurately!

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Math Problem Analysis

Mathematical Concepts

Exponential Functions
Coordinate Graphing

Formulas

f(x) = b^x (where b > 0 and b ≠ 1)
Properties of Exponents

Theorems

Exponential Growth Behavior

Suitable Grade Level

Grades 8-10

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