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

This task involves identifying the correct exponential function for each graph based on the curve’s behavior and key points provided. To assign the correct function, consider these elements for each graph:

  1. Exponential Growth or Decay: The direction of the curve (rising or falling) indicates whether the function is exponential growth (positive base greater than 1) or decay (base between 0 and 1).
  2. Vertical Intercept: The y-intercept gives a clue about the initial value of the function when x=0x = 0.
  3. Key Points: The points through which the curve passes help determine the function’s base and scale factor.

Let’s match each description with the appropriate function:

1. Graph 19:

  • Curve rises steeply in quadrant 3, crosses y-axis at (0,−1)(0, -1), then rises shallowly in quadrant 4.
  • Behavior: Exponential growth from negative values to approach the x-axis from below.
  • Likely function: y=−exy = -e^x.

2. Graph 20:

  • Curve rises shallowly in quadrant 2, crosses y-axis at (0,13)(0, \frac{1}{3}), then rises steeply in quadrant 1.
  • Behavior: Exponential growth with a fractional initial value.
  • Likely function: y=13exy = \frac{1}{3}e^x.

3. Graph 21:

  • Curve falls shallowly in quadrant 3, crosses y-axis at (0,−1)(0, -1), then falls steeply in quadrant 4.
  • Behavior: Exponential decay from negative values.
  • Likely function: y=−e−xy = -e^{-x}.

4. Graph 22:

  • Curve rises shallowly in quadrant 2, crosses y-axis at (0,1)(0, 1), then rises steeply in quadrant 1.
  • Behavior: Exponential growth with an initial value of 1.
  • Likely function: y=exy = e^x.

5. Graph 23:

  • Curve rises shallowly in quadrant 3, crosses y-axis at (0,0)(0, 0), then rises steeply in quadrant 1.
  • Behavior: Exponential growth starting from zero with an asymptote at y=−1y = -1.
  • Likely function: y=ex−1y = e^x - 1.

If you have the list of potential functions, align them with these interpretations. Let me know if further clarification or explanation is needed.

Follow-Up Questions:

  1. What is the general form of exponential functions?
  2. How does the base of an exponential function affect the graph?
  3. What role does the vertical intercept play in identifying exponential functions?
  4. How do exponential growth and decay differ visually?
  5. How can transformations like y=ex−1y = e^x - 1 affect the function’s graph?

Tip:

When analyzing exponential graphs, always look for the y-intercept and asymptotes, as they are key in distinguishing functions.

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Math Problem Analysis

Mathematical Concepts

Exponential Functions
Graph Analysis
Asymptotic Behavior
Growth and Decay

Formulas

y = a * e^x
y = a * e^(-x)
y = a * b^x + c

Theorems

Properties of Exponential Growth and Decay
Graph Transformations of Exponential Functions

Suitable Grade Level

Grades 9-12

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