Math Problem Statement

The graph of an exponential function is given. Select the function for each graph from the given options. Question content area bottom Part 1 Drag each function given above into the area below the appropriate​ graph, depending on which function is represented by which graph. 19. -3 3 -10 2 x y

A coordinate system has a horizontal x-axis labeled from negative 3 to 3 in increments of 0.5 and a vertical y-axis labeled from negative 10 to 2 in increments of 1. From left to right, a curve rises steeply in quadrant 3 passing through (negative 2, negative 9) and (negative 1, negative 3), crosses the y-axis at (0, negative 1), and then rises shallowly in quadrant 4 passing through approximately (1, negative one third). As x increases towards positive infinity, the curve approaches the positive x-axis from below. 20. -3 3 -2 10 x y

A coordinate system has a horizontal x-axis labeled from negative 3 to 3 in increments of 0.5 and a vertical y-axis labeled from negative 2 to 10 in increments of 1. From left to right, a curve rises shallowly in quadrant 2 passing through (negative 1, one ninth) and crosses the y-axis at (0, one third), and then rises steeply in quadrant 1 passing through approximately (1, 1); (2, 3); and (3, 9). As x decreases towards negative infinity, the curve approaches the negative x-axis from above. 21. -3 3 -10 2 x y

A coordinate system has a horizontal x-axis labeled from negative 3 to 3 in increments of 0.5 and a vertical y-axis labeled from negative 10 to 2 in increments of 1. From left to right, a curve falls shallowly in quadrant 3 passing through (negative 1, negative one third), crosses the y-axis at (0, negative 1), and then falls steeply in quadrant 4 passing through approximately (1, negative 3) and (2, negative 9). As x decreases towards negative infinity, the curve approaches the negative x-axis from below. 22. -3 3 -2 10 x y

A coordinate system has a horizontal x-axis labeled from negative 3 to 3 in increments of 0.5 and a vertical y-axis labeled from negative 2 to 10 in increments of 1. From left to right, a curve rises shallowly in quadrant 2 passing through (negative 1, one third), crosses the y-axis at (0, 1), and then rises steeply in quadrant 1 passing through approximately (1, 3) and (2, 9). As x decreases towards negative infinity, the curve approaches the negative x-axis from above. 23. -3 3 -2 10 x y

A coordinate system has a horizontal x-axis labeled from negative 3 to 3 in increments of 0.5 and a vertical y-axis labeled from negative 2 to 10 in increments of 1. From left to right, a curve rises shallowly in quadrant 3, crosses the y-axis at (0, 0), and then rises steeply in quadrant 1 passing through approximately (1, 2) and (2, 8). As x decreases towards negative infinity, the curve approaches the line y = negative 1 from above. 24. -3 3 -2 10 x y

Solution

To match each graph with its corresponding exponential function, let's analyze the characteristics of the graphs and compare them to the general behavior of exponential functions. Here's the general approach for each graph:

  1. Graph Analysis:

    • Identify key points on the curve.
    • Look for intercepts and the general shape (growth/decay, steep/shallow sections).
    • Note the asymptotic behavior (what happens as x→∞x \to \infty and x→−∞x \to -\infty).

  2. Function Behavior:

    • Exponential growth functions have the form y=a⋅bxy = a \cdot b^x with b>1b > 1.
    • Exponential decay functions have b<1b < 1.
    • Reflection across axes might involve negatives in the coefficient aa.

Detailed Matching of Graphs:

  1. Graph 19:

    • As xx increases, the curve approaches the x-axis from below.
    • Key points: (−2,−9),(−1,−3),(0,−1)(-2, -9), (-1, -3), (0, -1).
    • This is likely a reflected exponential growth function, such as y=−a⋅bxy = -a \cdot b^x.

  2. Graph 20:

    • The curve shows exponential growth with b>1b > 1.
    • Key points: (−1,19),(0,13),(1,1),(2,3),(3,9)(-1, \frac{1}{9}), (0, \frac{1}{3}), (1, 1), (2, 3), (3, 9).
    • Function form: y=13⋅bxy = \frac{1}{3} \cdot b^x, with b>1b > 1.

  3. Graph 21:

    • The curve resembles exponential decay, inverted about the x-axis.
    • Key points: (−1,−13),(0,−1),(1,−3),(2,−9)(-1, -\frac{1}{3}), (0, -1), (1, -3), (2, -9).
    • Function form: y=−a⋅bxy = -a \cdot b^x, with b>1b > 1.

  4. Graph 22:

    • This is exponential growth with an intercept at y=1y = 1.
    • Key points: (−1,13),(0,1),(1,3),(2,9)(-1, \frac{1}{3}), (0, 1), (1, 3), (2, 9).
    • Function form: y=a⋅bxy = a \cdot b^x, where a=1a = 1 and b>1b > 1.

  5. Graph 23:

    • The curve rises from quadrant 3, crosses through the origin (0,0)(0, 0), and exhibits steep growth.
    • Key points: (1,2),(2,8)(1, 2), (2, 8).
    • Function form: y=a⋅bx−1y = a \cdot b^x - 1 to reflect the asymptote at y=−1y = -1.

  6. Graph 24:

    • This graph is not fully described, but based on exponential patterns, it could involve variations similar to the others with intercepts and steep growth/decay patterns.

To finalize the function for each graph:

  1. What are the exact function options provided? List the options or clarify if you want explicit functions determined for each graph.

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Math Problem Analysis

Mathematical Concepts

Exponential Functions
Graph Interpretation
Asymptotic Behavior

Formulas

y = a * b^x (Exponential Growth)
y = -a * b^x (Reflected Exponential Growth)
y = a * b^x + c (Vertical Shift of Exponential Function)

Theorems

Exponential Growth and Decay Properties
Intercepts and Asymptotes of Exponential Functions

Suitable Grade Level

Grades 10-12

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