Absolute Value Functions Overview

Which of the following represents the vertex?

Which of the following is the graph of f(x) = |x| translated 2 units right, 2 units up, and dilated by a factor of?

What is the range of the absolute value function below?

Which statement is true about f(x) = -6|x + 5| - 2?

What is the vertex of the absolute value function below?

What is the domain of the absolute value function below?

Which function has a vertex at (2, 6)?

What is the vertex of f(x) = |x + 8| - 3?

Which of the following is the graph of f(x) = -0.5|x + 3| - 2?

Given the function f(x) = 0.5|x - 4| - 3, for what values of x is f(x) = 7?

Absolute Value Functions Overview

Absolute value functions take the form f(x) = a|x - h| + k, where (h, k) is the vertex.
The vertex represents the minimum or maximum point of the graph.

Transformation and Graphing

Translating the graph involves shifting it horizontally and vertically, as well as scaling.
A function f(x) = |x| can be transformed by:
- Moving 2 units right
- Moving 2 units up
- Dilation by a factor, affecting the steepness of the graph.

Range and Vertex Calculation

The range of absolute value functions is always [k, ∞) if the vertex is at (h, k).
To determine the vertex from a function, identify values of h and k that define the position on the graph.

Specific Function Characteristics

Function f(x) = -6|x + 5| - 2 exhibits horizontal compression due to the negative coefficient of absolute value.
The vertex for specific functions can often be calculated or identified through analysis, such as for f(x) = 2|x - 2| + 6, where the vertex is at (2, 6).

Domain Considerations

The domain of absolute value functions is typically all real numbers, represented as (-∞, ∞).

Problem Solving with Function Values

For complex absolute value functions, such as f(x) = 0.5|x - 4| - 3, solutions for specific output values can be found by setting the function equal to the desired outcome and solving for x.

This quiz covers the key concepts of absolute value functions, including their form, transformations, and graphing techniques. You will learn how to identify the vertex and range of these functions, as well as their specific characteristics. Test your understanding of these essential algebra concepts.