Math Problem Statement

Complete an equation for the function graphed above.

Solution

The graph in the image represents a "V"-shaped graph, which suggests it is the graph of an absolute value function. The function seems to have a vertex at (1,1)(-1, -1), and the slopes of the two linear parts are different on either side of the vertex. Let's break this down.

Key observations:

  1. Vertex: The point (1,1)(-1, -1) is the lowest point, indicating that this is the vertex of the function.
  2. Slope:
    • On the left of x=1x = -1 (from (5,3)(-5, 3) to (1,1)(-1, -1)), the slope is 44=1\frac{-4}{4} = -1.
    • On the right of x=1x = -1 (from (1,1)(-1, -1) to (3,3)(3, 3)), the slope is 44=1\frac{4}{4} = 1.

General form of an absolute value function:

y=axh+ky = a|x - h| + k Where:

  • (h,k)(h, k) is the vertex, so h=1h = -1 and k=1k = -1.
  • The slope aa is determined by the rate of change on either side of the vertex. Here, a=1a = 1 since the slopes are 11 and 1-1.

Therefore, the equation of the function is:

y=x+11y = |x + 1| - 1

Let me know if you need further clarification or details!


Here are five questions that build on this:

  1. What is the significance of the vertex in an absolute value function?
  2. How would the graph shift if we changed the equation to y=x+21y = |x + 2| - 1?
  3. How does the slope change affect the shape of the graph?
  4. What would the equation be if the vertex were located at (2,3)(-2, 3)?
  5. How do you interpret the impact of the constant kk in the absolute value function equation?

Tip: Always identify the vertex when working with absolute value functions—it helps set the foundation for the equation.

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Math Problem Analysis

Mathematical Concepts

Algebra
Absolute Value Functions
Graphing

Formulas

y = a|x - h| + k

Theorems

Properties of Absolute Value Functions

Suitable Grade Level

Grades 8-10