Math Problem Statement

What are the coordinates of the vertex of the graph of f(x) = 4|x + 6|?

Solution

To find the vertex of the function f(x)=4x+6f(x) = 4|x + 6|, we analyze the form of an absolute value function.

  1. The function is given by: f(x)=4x+6f(x) = 4|x + 6|

  2. The general form of an absolute value function is: f(x)=axh+kf(x) = a |x - h| + k where (h,k)(h, k) represents the vertex of the function.

  3. In f(x)=4x+6f(x) = 4|x + 6|, we can rewrite it as: f(x)=4x(6)f(x) = 4 |x - (-6)| This shows that h=6h = -6 and k=0k = 0.

  4. Therefore, the vertex of this absolute value function is located at: (6,0)(-6, 0)

Answer: The coordinates of the vertex are (6,0)(-6, 0).

Would you like further explanation on this topic or have any questions?

Related Questions

  1. How do you determine the vertex of a general absolute value function?
  2. What effect does changing the coefficient of the absolute value term have on the graph?
  3. How would the graph shift if we added a constant term to f(x)=4x+6f(x) = 4|x + 6|?
  4. What would the graph look like if the absolute value term had a negative coefficient?
  5. How does the slope of the lines formed in an absolute value graph relate to the coefficient outside the absolute value?

Tip:

Remember, the vertex of an absolute value function f(x)=axh+kf(x) = a |x - h| + k is always at the point (h,k)(h, k), where hh shifts the graph horizontally and kk shifts it vertically.

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Math Problem Analysis

Mathematical Concepts

Algebra
Absolute Value Functions
Vertex Form of Absolute Value Functions

Formulas

Vertex form of an absolute value function f(x) = a|x - h| + k

Theorems

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Suitable Grade Level

Grades 7-10