Math Problem Statement

Find each key feature of the function shown in the graph. Enter all intervals using interval notation. Enter the x-intercept as values of x only.

The graph with X-coordinate marks -4, -2, 0, 2, 4 and Y-coordinate mark -4, -2, -1, 0, 2 and 4. There are 3 key features. The first key feature has head at (0, 1) and tail at (-1, 0). The second key feature has head at (1, 0) and tail at (0, -1). The third key feature has head at (2, -1) and tail at (1, -2).

range =

domain =

x-intercept(s) =

average rate of change over the interval [0, 1] =

an interval where the graph is positive =

Solution

The graph shows three segments, and the key features are as follows:

  1. Range: The range refers to all possible yy-values that the function can take. From the graph, the yy-values range from 2-2 to 11. Therefore, the range is: [2,1][-2, 1]

  2. Domain: The domain refers to all possible xx-values that the function can take. From the graph, the xx-values range from 1-1 to 22. Therefore, the domain is: [1,2][-1, 2]

  3. X-intercepts: The x-intercepts are the points where the graph crosses the xx-axis. From the graph, the x-intercepts are at x=1x = -1 and x=1x = 1. Therefore, the x-intercepts are: x=1,1x = -1, 1

  4. Average Rate of Change Over the Interval [0,1][0, 1]: The average rate of change is given by the formula: Average Rate of Change=f(b)f(a)ba\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} For the interval [0,1][0, 1]:

    • f(0)=1f(0) = 1
    • f(1)=0f(1) = 0 Therefore, the average rate of change over [0,1][0, 1] is: 0110=11=1\frac{0 - 1}{1 - 0} = \frac{-1}{1} = -1
  5. An Interval Where the Graph Is Positive: The graph is positive where the function is above the x-axis. From the graph, the interval where the function is positive is: (1,0)(-1, 0)

If you have any questions or need further explanations, feel free to ask!

Here are 5 related questions to deepen your understanding:

  1. How do you determine the intervals where a function is increasing or decreasing?
  2. What is the significance of the slope in determining the rate of change?
  3. Can a function have more than one x-intercept? What does it mean?
  4. How would you calculate the average rate of change over a different interval, such as [1,2][1, 2]?
  5. How do you find the y-intercept from a graph, and what does it represent?

Tip: Remember, the domain and range of a function provide crucial information about the limits of xx and yy-values, helping you understand the behavior of the function across its entire graph.

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Math Problem Analysis

Mathematical Concepts

Functions
Graphing
Interval Notation
Average Rate of Change

Formulas

Average Rate of Change formula: (f(b) - f(a)) / (b - a)

Theorems

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Suitable Grade Level

Grades 9-12