Absolute Value Function

Definition

The absolute value function is a mathematical function that gives the distance of a real number from zero on the number line. It is denoted by the symbol |x| and represents the positive value of the input x, regardless of its sign.

5 Must Know Facts For Your Next Test

1. The absolute value function is an even function, meaning it is symmetric about the y-axis.
2. The graph of the absolute value function is a V-shaped curve that opens upward and has a vertex at the origin.
3. The absolute value function is a piecewise function, with the formula $|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$
4. The absolute value function has a domain of all real numbers and a range of non-negative real numbers.
5. Transformations of the absolute value function, such as shifts, reflections, and dilations, can be used to create a variety of absolute value graphs.

Review Questions

• Explain the geometric interpretation of the absolute value function and how it relates to the distance of a number from zero on the number line.
  • The absolute value function represents the distance of a number from zero on the number line. Geometrically, the graph of the absolute value function is a V-shaped curve that opens upward and has a vertex at the origin. The absolute value of a number $x$ is the distance between $x$ and 0 on the number line, regardless of whether $x$ is positive or negative. This means that the absolute value function always gives a non-negative output, as it represents the positive distance from zero.
• Describe the relationship between the absolute value function and piecewise functions. How does this relationship affect the graph and properties of the absolute value function?
  • The absolute value function is a specific type of piecewise function. The formula for the absolute value function is a piecewise definition, where the function is defined differently for positive and negative inputs. This piecewise nature of the absolute value function affects its graph and properties. The graph of the absolute value function is a V-shaped curve, with the two linear pieces meeting at the vertex (the origin). Additionally, the piecewise definition contributes to the absolute value function being an even function, meaning it is symmetric about the y-axis.
• Analyze how transformations, such as shifts, reflections, and dilations, can be applied to the absolute value function to create a variety of absolute value graphs. Explain the effect of these transformations on the function's properties.
  • Transformations of the absolute value function can be used to create a wide range of absolute value graphs. Shifting the function horizontally or vertically will change the location of the vertex, but the V-shaped curve and the even function property will be preserved. Reflecting the function across the x-axis or y-axis will change the sign of the function's output, which can be interpreted as changing the direction of the distance from zero. Dilating the function will change the steepness of the V-shaped curve, affecting the rate of change and the function's sensitivity to input values. These transformations allow for the creation of a variety of absolute value graphs while maintaining the core characteristics of the absolute value function.

Related terms

Absolute Value: The absolute value of a number is its distance from zero on the number line. It is always a non-negative number.

Piecewise Function: A piecewise function is a function that is defined by different formulas or expressions on different intervals of its domain.

Reflection: The reflection of a function across the x-axis or y-axis, which changes the sign of the function's output.