Absolute Value Functions
Getting to Know Absolute Value Functions
Absolute Value Functions are equations with absolute value bars, so when solved, they have two solutions. When graphed, it looks like a "v."
Parent function of an absolute value function is y=|x|
The Domain of an absolute value function is all of the possible solutions for x, or how far left or right the graph goes.
The Range of an absolute value function is all of the possible solutions for y, or how far up or down the graph goes.
If you graphed the parent function, the domain would be all real numbers and the range would be anything more than or equal to zero.
Parent function of an absolute value function is y=|x|
The Domain of an absolute value function is all of the possible solutions for x, or how far left or right the graph goes.
The Range of an absolute value function is all of the possible solutions for y, or how far up or down the graph goes.
If you graphed the parent function, the domain would be all real numbers and the range would be anything more than or equal to zero.
Standard Form
The standard, or simplified, form of an absolute value function is y= |mx+b| +c
The vertex of this formula can be found by plugging numbers into this: (-b/2a, c). This way, the axis of symmetry is equal to -b/2a. When x is zero, the y axis will be the axis of symmetry.
This form is used to find the graph. It is important that it fits into this, because it will give you all of the right values for m, b, and c. If you do not have the right values, then the answer will be wrong.
The vertex of this formula can be found by plugging numbers into this: (-b/2a, c). This way, the axis of symmetry is equal to -b/2a. When x is zero, the y axis will be the axis of symmetry.
This form is used to find the graph. It is important that it fits into this, because it will give you all of the right values for m, b, and c. If you do not have the right values, then the answer will be wrong.
Vertex Form
The vertex form of an absolute value function is y= a|x-h| +k
This form is used to find the vertex of a graph easily. (h,k) is the point of the vertex, so all you have to do to find the vertex is plug in the h and k values from the equation.
This form is used to find the vertex of a graph easily. (h,k) is the point of the vertex, so all you have to do to find the vertex is plug in the h and k values from the equation.
Types of Translations
Not all the time is the vertex the origin. Horizontal and vertical shifts can change this. Dilations and reflections can also change the way your graph looks.
Horizontal translation: The "h" in vertex form determines how far left or right the vertex moves. If the number is negative, you move it right however many units. If the number is positive, you move it to the left however many units.
Vertical translation: The "k" in the vertex form determines how far up or down the vertex moves. If the number is negative, you move it down however many units. If the number is positive you move it up however many units.
Dilation: The "a" value determines how wide or skinny the "v" is. If the value is more than one, then the parabola will be skinnier than the graphed parent function. If the value is more than zero but less than one, then the "v" will be wider than the graphed parent function.
Horizontal translation: The "h" in vertex form determines how far left or right the vertex moves. If the number is negative, you move it right however many units. If the number is positive, you move it to the left however many units.
Vertical translation: The "k" in the vertex form determines how far up or down the vertex moves. If the number is negative, you move it down however many units. If the number is positive you move it up however many units.
Dilation: The "a" value determines how wide or skinny the "v" is. If the value is more than one, then the parabola will be skinnier than the graphed parent function. If the value is more than zero but less than one, then the "v" will be wider than the graphed parent function.
Reflections: the "a" in vertex form determines if the "v" is reflected over the x-axis or not. If there is a negative sign in front of it, then it will be reflected over the x-axis.
*This video will go over all of the translations and better explain them. It refers to dilations as stretches and shrinks.
*This video will go over all of the translations and better explain them. It refers to dilations as stretches and shrinks.
Notable Points
An absolute value equation is more than the equation and the line. It also includes key terms like the axis of symmetry and the vertex.
Axis of symmetry: This splits a graph in two mirror images. It goes through the vertex. The value of x determines where the axis of symmetry is. If x is 4 in the vertex, then x=4 is the axis of symmetry.
Vertex: The vertex is the highest or lowest point of the graph. The axis of symmetry goes through this point.
* In this image, the axis of symmetry is the y-axis. The vertex, is as shown, the origin.
Axis of symmetry: This splits a graph in two mirror images. It goes through the vertex. The value of x determines where the axis of symmetry is. If x is 4 in the vertex, then x=4 is the axis of symmetry.
Vertex: The vertex is the highest or lowest point of the graph. The axis of symmetry goes through this point.
* In this image, the axis of symmetry is the y-axis. The vertex, is as shown, the origin.
How to Graph Absolute Value Inequalities
*This video will teach you how to graph an absolute value inequality. It will also demonstrate how to find the equation of an absolute inequality graph.