Absolute Value Function

Definition

The absolute value function is a mathematical function that describes the distance of a number from zero on the number line. It represents the magnitude or size of a number, regardless of its sign. The absolute value of a number is always a non-negative value.

5 Must Know Facts For Your Next Test

1. The absolute value of a number is denoted by the vertical bars surrounding the number, such as |x|.
2. The absolute value function is always non-negative, meaning the output will never be a negative number.
3. The graph of an absolute value function is a V-shaped curve that opens upward or downward, depending on the function.
4. Transformations of absolute value functions can be used to model a variety of real-world situations, such as distance, height, and financial data.
5. Absolute inequalities can be solved by considering the positive and negative solutions that satisfy the inequality.

Review Questions

• Explain how the absolute value function is used to represent the distance of a number from zero on the number line.
  • The absolute value function |x| represents the distance of the number x from the origin (0) on the number line. This distance is always a non-negative value, regardless of whether x is positive or negative. For example, |5| = 5 and |-3| = 3, as both 5 and 3 represent the distance from 0 on the number line. The absolute value function is a way to describe the magnitude or size of a number without regard to its sign.
• Describe the key features of the graph of an absolute value function and how they are affected by transformations.
  • The graph of an absolute value function $f(x) = |x|$ is a V-shaped curve that opens upward. The vertex of the V is located at the origin (0,0), and the graph passes through the points (0,0), (-1,-1), and (1,1). Transformations of the absolute value function can alter the shape, position, and orientation of the graph. For example, a vertical stretch by a factor of 2 would result in the graph $f(x) = 2|x|$, which is a narrower V-shape. A horizontal shift to the right by 3 units would give the graph $f(x) = |x-3|$, which is shifted to the right. These transformations allow absolute value functions to model a variety of real-world situations involving distance, height, and other measurable quantities.
• Explain how to solve absolute value inequalities and discuss the significance of the positive and negative solutions.
  • Solving absolute value inequalities, such as $|x-2| < 4$, requires considering both the positive and negative solutions that satisfy the inequality. To solve this inequality, we can rewrite it as two separate inequalities: $x-2 < 4$ and $-(x-2) < 4$. Solving these inequalities gives the solutions $-2 < x < 6$. The significance of the positive and negative solutions is that they represent the range of values for the variable x that satisfy the absolute value inequality. In this example, the absolute value inequality is satisfied by all values of x between -2 and 6, including both positive and negative numbers. Understanding how to solve absolute value inequalities and interpret the solutions is crucial for modeling real-world situations involving constraints or thresholds.

Related terms

Graphing Absolute Value Functions: Graphing absolute value functions involves plotting the distance of a number from the origin (0,0) on the coordinate plane. The resulting graph is a V-shaped curve that opens upward or downward.

Transformations of Absolute Value Functions: Absolute value functions can be transformed by shifting, stretching, or reflecting the graph. These transformations include translations, reflections, and dilations, which can be applied to the input, output, or both.

Absolute Inequality: Absolute inequalities involve comparing the absolute value of an expression to a constant value. Solving absolute inequalities requires considering both the positive and negative solutions that satisfy the inequality.