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Miscellaneous Functions

Analyzing Graphs of Absolute Value Functions

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Concept

Absolute Value Function

An absolute value function is a function that contains an absolute value expression.

An absolute value graph has a distinct V-shape. It is symmetric about the vertical line that passes through its vertex. The vertex of an absolute value function is the point where the graph changes direction.

Graph of the absolute value function y=|x-1|+1 which opens upwards and has a vertex at (1,1).
Since the graph of y=x1∣+1 changes direction at (1,1), this point is the vertex. Additionally, the graph is symmetric about the line x=1.

Example

Graph the absolute value function

fullscreen

Graph the absolute value function f(x)=x2∣3 using a table of values.

Show Solution expand_more
To graph the given function using a table of values, we can substitute various x-values into the rule and solve for the corresponding y-values. The absolute value of a number is always the positive value of that number. For instance, -2∣=2. Let's first calculate the y-value that corresponds with x=-3.
f(x)=x2∣3
f(-3)=-32∣3
f(-3)=-5∣3
f(-3)=53
f(-3)=2
We have found that f(-3)=2. Thus, the point (-3,2) lies on the graph. We can find other points on the graph in the same way.
x x2∣3 f(x)
-2 -22∣3 1
-1 -12∣3 0
0 02∣3 -1
1 12∣3 -2
2 22∣3 -3
3 32∣3 -2
4 42∣3 -1
5 52∣3 0
6 62∣3 1
7 72∣3 2

To draw the graph, we can plot these points, then connect them with a V-shaped curve.

Method

Solving Absolute Value Equations Graphically

An absolute value equation is an equation that contains the absolute value of a variable expression. An example of this kind of equation is
As is the case with most equations, these can be solved graphically. This is done by moving all terms except the constant term to one side. The function, which is the non-constant side of the equation, is then graphed and the solution(s) to the equation are found as the point(s) on the graph having the y-coordinate that equals the constant.

Example

Solve the absolute value equation graphically

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Solve the equation graphically.
∣2x5∣3=0
Show Solution expand_more
When solving an equation graphically, the first step is to rearrange the equation so that the constant term is alone on one side.
The left-hand side of the equation can now be expressed as a function, f(x)=∣2x5∣. We'll draw the graph of f.
The solutions to the equation
y=∣2x5∣=3
are the x-coordinates of the points on the graph that have the y-coordinate 3.
The equation has the solutions x=1 and x=4. We can verify the solutions by testing them in the equation. We'll start with x=1.
∣2x5∣=3
3=3
Since x=1 makes a true statement, it is a solution to the equation. Next, we'll verify x=4 in the same way.
Since x=4 makes a true statement, it is also a solution. Thus, the equation has the solutions
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