How do you find the vertex of an absolute value function?

The vertex of an absolute value function, generally in the form f(x) = a |x - h| + k, is found at the point (h, k).

Understanding the Absolute Value Function

The standard form of an absolute value function is:

f(x) = a |x - h| + k

Where:

• a affects the stretch or compression and reflection of the graph.
• (h, k) represents the vertex of the absolute value graph.
• |x - h| represents the absolute value of the expression.

Methods to Find the Vertex

There are two primary ways to determine the vertex:

1. Using the Standard Form

• Identify h and k: Directly observe the values of h and k in the equation f(x) = a |x - h| + k. Remember that the value of h is the opposite sign of what appears inside the absolute value.
• Vertex coordinates: The vertex is then simply (h, k).

Example:

Consider the function f(x) = 2 |x - 3| + 4. Here, h = 3 and k = 4. Therefore, the vertex is (3, 4).

2. Finding the Zero of the Absolute Value

• Set the expression inside the absolute value to zero: Solve the equation x - h = 0 for x. This will give you the x-coordinate of the vertex.
• Substitute to find k: Substitute the x-value you just found back into the original function, f(x) = a |x - h| + k. The result will be the y-coordinate, k, of the vertex.

Example:

Consider the function f(x) = -|x + 1| + 5.

1. Set x + 1 = 0. Solving for x, we get x = -1.
2. Substitute x = -1 into the function: f(-1) = -|-1 + 1| + 5 = -|0| + 5 = 5.

Therefore, the vertex is (-1, 5).

Key Considerations

• The vertex represents the minimum point of the graph if a > 0 and the maximum point if a < 0.
• The absolute value function creates a V-shaped graph, and the vertex is the sharp corner of the V.

By using either method, you can accurately determine the vertex of any absolute value function in the given form.