Math Problem Statement

Use the graph to the left to answer questions 18-21.

Solution

Let's work through the graph together based on the image provided. Here's a breakdown of each question:

18. What are the x-intercepts of the graph?

From the graph, the x-intercepts are the points where the graph crosses the x-axis. The graph crosses the x-axis at two points:

  • x=2x = -2
  • x=2x = 2

19. Increasing:

The graph is increasing on the interval where it moves upward as you move from left to right. This happens between x=2x = -2 and x=0x = 0. Therefore, the function is increasing on the interval:

  • (2,0)(-2, 0)

20. Decreasing:

The graph is decreasing where it moves downward as you move from left to right. This occurs in two sections:

  • For x<2x < -2, meaning (,2)(-\infty, -2)
  • For x>0x > 0, meaning (0,)(0, \infty)

21. End Behavior:

End behavior describes how the function behaves as xx approaches positive and negative infinity.

  • As xx \to -\infty, f(x)f(x) \to \infty (the graph rises towards infinity).
  • As xx \to \infty, f(x)f(x) \to \infty (the graph also rises towards infinity).

The graph seems to resemble a downward-facing parabola that flips, indicating the described end behavior.


Let me know if you have any questions about the explanation or want additional details. Here are five relative questions for further exploration:

  1. What is the vertex of this parabola, and how is it related to the intervals of increase and decrease?
  2. What is the axis of symmetry for this graph?
  3. How would you find the y-intercept of the graph from the given information?
  4. Can you write an equation that models this parabola using its intercepts?
  5. What is the domain and range of the function shown in the graph?

Tip:

When analyzing the end behavior of a graph, always look at the direction the graph heads as xx \to \infty and xx \to -\infty to understand how the function grows or decreases.

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Math Problem Analysis

Mathematical Concepts

Graph Analysis
Quadratic Functions
Intervals
End Behavior

Formulas

Standard quadratic equation: ax^2 + bx + c = 0

Theorems

Behavior of quadratic functions
Properties of parabolas

Suitable Grade Level

Grades 9-11