Math Problem Statement

The illustration below shows the graph of y as a function of x. Complete the following sentences based on the graph of the function.

Solution

I see the graph and accompanying sentences. Let's work through it step by step:

  1. Initially, as xx increases, yy decreases.

    • From the graph, between x=0x = 0 and x=2x = 2, the value of yy decreases from y=0y = 0 to y=2y = -2.
  2. The slope of the graph is equal to -1 for all xx between x=0x = 0 and x=3x = 3.

    • From x=0x = 0 to x=3x = 3, the graph decreases linearly with a slope of -1 (the line moves down by 1 unit for each 1 unit increase in xx).
  3. Starting at x=3x = 3, the function value yy increases as xx increases.

    • After x=3x = 3, the function rises from y=3y = -3 to y=2y = 2 at x=5x = 5, so the function value increases.
  4. The slope of the graph is equal to 2 for xx between x=3x = 3 and x=5x = 5.

    • Between x=3x = 3 and x=5x = 5, the slope is 2 (the line goes up 2 units for every 1 unit increase in xx).
  5. For xx between x=0x = 0 and x=4x = 4, the function value yy is less than 0.

    • Between x=0x = 0 and x=4x = 4, the function value yy is below the x-axis, meaning it is less than 0.
  6. For xx between x=4x = 4 and x=8x = 8, the function value yy is greater than 0.

    • After x=4x = 4, the function rises above the x-axis, making yy positive.

Would you like to go over any part of this, or would you like further details on specific aspects?

Here are five related questions to expand on:

  1. How do you calculate the slope of a line between two points?
  2. What does it mean for a function to increase or decrease in value?
  3. How can you tell if a function is linear from its graph?
  4. What is the importance of identifying when a function crosses the x-axis?
  5. Can you describe the difference between local maxima and minima in the graph?

Tip: The slope of a line segment in a graph tells you how steep the line is and in what direction (positive or negative).

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Math Problem Analysis

Mathematical Concepts

Linear Functions
Slope
Graph Interpretation

Formulas

Slope formula: (y2 - y1) / (x2 - x1)

Theorems

Properties of linear functions
Concept of increasing and decreasing functions

Suitable Grade Level

Grades 8-10