The parent function of y absolute value functions is represented by the equation y = |x|, where the absolute value of a number x is denoted by |x|. This is the simplest form of an absolute value function, and it serves as a reference from which other absolute value functions can be derived.
What does the parent function y = |x| represent?
The parent function y = |x| represents the absolute value of the variable x. It calculates the distance of x from the origin on a number line.
What is an absolute value?
An absolute value is a non-negative value that represents the magnitude or distance of a number from zero. It is denoted by the symbol |x|.
What is the shape of the graph of the parent function y = |x|?
The graph of the parent function y = |x| is in the shape of a “V” or a “V” turned upside down. It is symmetrical about the y-axis and passes through the origin (0, 0).
What is the domain and range of the parent function?
The domain of the parent function y = |x| is all real numbers, as any value of x can be substituted into the equation. The range, however, is limited to non-negative real numbers (including zero), as the absolute value is always positive or zero.
How can the parent function be modified to shift its graph?
To shift the graph of the parent function, you can add a constant value to the x or y variable. For example, y = |x| + 2 would shift the graph up by 2 units, while y = |x – 3| would shift the graph to the right by 3 units.
What happens when the coefficient of x is negative in the parent function?
When the coefficient of x is negative, the graph of the absolute value function reflects vertically over the x-axis. For instance, y = -|x| has a “V” shape opening downwards.
Can the parent function be stretched or compressed?
No, the parent function y = |x| cannot be stretched or compressed. It maintains a fixed slope of ±1.
How does changing the coefficient affect the parent function?
Changing the coefficient of x in the parent function stretches or compresses the graph vertically. For example, y = 2|x| would stretch the graph vertically by a factor of 2.
What are some real-world applications of absolute value functions?
Absolute value functions find applications in various fields, such as physics, engineering, economics, and computer science. They are used to model situations where distance, magnitude, or efficiency matters, such as calculating speed, determining optimal pricing, or analyzing waveforms.
Can absolute value functions intersect the x-axis?
No, absolute value functions cannot intersect the x-axis because they always maintain a non-negative value or zero due to the absolute value property.
Is the parent function symmetric?
Yes, the parent function y = |x| is symmetric about the y-axis. This means that if (x, y) is a point on the graph, then (-x, y) would also be a point on the graph.
Can the graph of an absolute value function have more than one bend?
No, the graph of an absolute value function can have only one bend or vertex. Its shape can either be a “V” or a “V” turned upside down, depending on the coefficients and constants in the equation.
What happens if there are multiple absolute values in an equation?
When there are multiple absolute values in an equation, the graph will have different pieces that may or may not intersect. The equation can be divided into intervals, with each interval having a different equation to represent it.
How can translations affect the parent function?
Translations involve shifting the graph horizontally or vertically. Horizontal translations (shifts) will influence the x-value in the equation, while vertical translations affect the y-value. These shifts change the position of the graph without altering its shape.