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Miscellaneous Functions

Analyzing Graphs of Absolute Value Functions

Concept

Absolute Value Function

An absolute value function is any function that contains the absolute value of a variable expression. These functions can also be described as any function that is a transformation of the absolute value parent function,
f(x)=x∣.
Since the absolute value of an expression is never negative, the graph of the function f(x)=x will always lie on or above the x-axis. Note that
This means that the points (-5,5) and (5,5) both lie on f(x)=x∣. As it turns out, these points lie directly across from each other. In fact, this symmetry exists for all inverse input values. Thus, absolute value graphs have a distinct V-shape.
Any function belonging to the absolute value function family can be written using the equation
y=axh+k
where a, h, and k are real numbers and a0.
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Exercise

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

Show Solution
Solution
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.
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Exercise
Solve the equation graphically.
∣2x5∣3=0
Show Solution
Solution
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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