Absolute value functions are like mathematical boomerangs. They always bounce back, creating a V-shaped graph that's around a central point. These functions have unique properties that make them useful for modeling real-world situations.
Understanding absolute value functions unlocks a world of practical applications. From calculating distances to analyzing financial data, these functions help us make sense of situations where the magnitude matters more than the direction. They're essential tools in your mathematical toolkit.
Absolute Value Functions
Key features of absolute value graphs
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The of an is f(x)=a∣x−h∣+k (also known as the )
a determines the or compression of the graph
If ∣a∣>1, the graph is vertically stretched (appears steeper)
If 0<∣a∣<1, the graph is vertically compressed (appears flatter)
If a<0, the graph opens downward ()
h represents the of the
If h>0, the graph shifts to the right (positive direction)
If h<0, the graph shifts to the left (negative direction)
k represents the of the vertex
If k>0, the graph shifts up (positive direction)
If k<0, the graph shifts down (negative direction)
The graph of an absolute value function is V-shaped with a vertex at (h,k)
The vertex is the point where the graph changes direction
The graph is symmetric about the vertical line passing through the vertex
The of an absolute value function is all
The function is defined for any input value (x can be any real number)
The of an absolute value function is y≥k if a>0, or y≤k if a<0
If the graph opens upward, the range is all y values greater than or equal to the vertex's y-coordinate
If the graph opens downward, the range is all y values less than or equal to the vertex's y-coordinate
Transformations and Parent Function
The of absolute value is f(x)=[∣x∣](https://www.fiveableKeyTerm:∣x∣)
of the parent function include:
Vertical and horizontal shifts
Vertical and horizontal stretches or compressions
Reflections over the x-axis or y-axis
The absolute value function is continuous for all real numbers
Solving absolute value equations
To solve an , isolate the absolute value term on one side of the equation
If the other side is positive, split the equation into two and solve each part
Example: ∣x−3∣=5 becomes x−3=5 or x−3=−5
Solve x−3=5 by adding 3 to both sides: x=8
Solve x−3=−5 by adding 3 to both sides: x=−2
The solution set is {8,−2}
If the other side is negative, there is no solution
Example: ∣x+1∣=−4 has no solution because the absolute value is
When solving absolute value equations with variables on both sides, isolate one absolute value term and then follow the above steps
Example: ∣2x−1∣+3=∣x+2∣ becomes ∣2x−1∣=∣x+2∣−3
Simplify the right side: ∣2x−1∣=∣x+2∣−3
Split into two equations and solve each part
Absolute value functions can also be expressed using a
Real-world applications of absolute value
Absolute value can represent the distance between two points on a number line
Example: The distance between 3 and -5 is ∣3−(−5)∣=∣3+5∣=8
This concept can be applied to find the distance between any two points (locations, temperatures)
Absolute value can model situations where the direction is irrelevant, but the magnitude is important
Example: A company's profit or loss can be represented using absolute value, as the magnitude is more important than whether it's a profit (positive) or loss (negative)
If a company's profit/loss is represented by x, then ∣x∣ gives the magnitude of the profit/loss
Absolute value can be used to find the minimum or maximum distance between two functions
Example: To find the minimum distance between f(x)=x2 and g(x)=−x+2, set up an absolute value equation: ∣x2−(−x+2)∣=d
Solve for x when the distance d is minimized (vertex of the absolute value graph)
This technique can be used to optimize distances in various applications (network analysis, resource allocation)