Math Problem Statement

Simply

The function has a minimum value, and we need to find where it is increasing or decreasing based on the graph.

Solution

Let's analyze the graph in the image and answer the questions step by step.

1. Identifying the minimum of the function:

From the graph, we see that the curve reaches its lowest point at (x,y)=(4,4)(x, y) = (-4, -4), indicating that the function has a minimum value of -4 at x=4x = -4.

2. Identifying where the function is increasing:

The function increases when moving from left to right, as the graph rises. Looking at the graph:

  • The function increases from x=4x = -4 to x=0x = 0.

Thus, the function is increasing on the interval (4,0)(-4, 0).

3. Identifying where the function is decreasing:

The function decreases when the graph is falling. Looking at the graph:

  • The function decreases from x=6x = -6 to x=4x = -4.
  • It also decreases from x=0x = 0 to x=2x = 2.

Thus, the function is decreasing on the intervals (6,4)(-6, -4) and (0,2)(0, 2).


Summary:

  • The function has a minimum of 4-4 at x=4x = -4.
  • The function is increasing on the interval (4,0)(-4, 0).
  • The function is decreasing on the intervals (6,4)(-6, -4) and (0,2)(0, 2).

Would you like more details or have any questions about this? Here are some related questions you might want to explore:

  1. How do you determine where a function is increasing or decreasing from a graph?
  2. What does the derivative tell you about increasing and decreasing functions?
  3. How do you find critical points of a function algebraically?
  4. Can a function have more than one minimum or maximum point?
  5. How would you express the behavior of the function as xx approaches infinity?

Tip: When analyzing a graph, always look at the direction of the slope to determine increasing and decreasing intervals.

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Math Problem Analysis

Mathematical Concepts

Functions
Increasing and Decreasing Intervals
Quadratic Functions

Formulas

-

Theorems

Properties of Quadratic Functions
Critical Points and Derivatives

Suitable Grade Level

Grades 8-10