Technology
How to Draw the Modulus Graph of y |3x - 2x|
How to Draw the Modulus Graph of ( y |3x - 2x| )
Understanding the Function
The function we are dealing with is ( y |3x - 2x| ). This involves an absolute value, which means we need to consider where the expression inside the absolute value changes sign. Let's break down this process step by step.
Step 1: Understanding the Function
The function ( y 3x - 2x ) can be simplified to ( y x ) within the absolute value. However, the absolute value will change the way the function behaves depending on the value of ( x ). The absolute value function, ( |A| ), is defined as:
( |A| A ) if ( A geq 0 ) ( |A| -A ) if ( ATherefore, ( y |3x - 2x| |x| ).
Step 2: Identify Critical Points
The critical point is where the expression inside the absolute value is zero. This is found by solving:
[ 3x - 2x 0 implies x 0 ]
So, ( x 0 ) is the critical point where the behavior of the function changes.
Step 3: Analyze the Function in Different Intervals
We will analyze the function in the intervals ( (-infty, 0) ) and ( [0, infty) ).
For ( x [ 3x - 2x x implies y |x| -x ] For ( x geq 0 ): [ 3x - 2x x implies y |x| x ]Step 4: Find Points of Interest
Let's calculate the points of interest:
The Y-intercept: [ y |x| text{ at } x 0 implies y 0 ] The X-intercept: [ |x| 0 implies x 0 ]Step 5: Sketch the Graph
Based on the points of interest, we can sketch the graph:
Plot the Y-intercept at ( (0, 0) ). Plot the graph for ( x Plot the graph for ( x geq 0 ) as the line ( y x ). Since ( |x| ) is continuous at ( x 0 ), the graph will touch the x-axis at ( (0, 0) ).The graph will be a straight line increasing in the first quadrant and decreasing in the negative quadrant, meeting at the point ( (0, 0) ).
Conclusion
The graph of ( y |3x - 2x| ) is a V-shaped graph, with the vertex at the origin. The graph is continuous and reflects the behavior of the absolute value function.
Graphing Tool
You can use graphing software or a graphing calculator to visualize the function accurately. Here is a rough sketch of the graph:
Graphic representation of the modulus graph for y |3x - 2x|.
Feel free to ask if you need further assistance!
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