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What is the Graph of y = |x|?
Grade Level:
Class 9
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
The graph of y = |x| is a V-shaped graph that shows how the absolute value of a number (x) changes its output (y). It always gives a positive value for y, no matter if x is positive or negative. The graph is symmetrical about the y-axis, meaning it looks the same on both sides.
Simple Example
Quick Example
Imagine you are tracking how far a cricket ball travels from the bowler, whether it goes towards the leg side (negative direction) or off side (positive direction). If the ball travels 5 meters towards the leg side (-5m) or 5 meters towards the off side (+5m), the 'distance from the bowler' is always 5 meters. This 'distance' is like y = |x|, always positive.
Worked Example
Step-by-Step
Let's plot some points for y = |x|.
Step 1: Choose some values for x.
Let's pick x = -3, -2, -1, 0, 1, 2, 3.
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Step 2: Calculate the corresponding y values using y = |x|.
For x = -3, y = |-3| = 3.
For x = -2, y = |-2| = 2.
For x = -1, y = |-1| = 1.
For x = 0, y = |0| = 0.
For x = 1, y = |1| = 1.
For x = 2, y = |2| = 2.
For x = 3, y = |3| = 3.
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Step 3: List the (x, y) coordinate pairs.
(-3, 3), (-2, 2), (-1, 1), (0, 0), (1, 1), (2, 2), (3, 3).
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Step 4: Plot these points on a graph paper.
Plot each pair of coordinates on the x-y plane.
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Step 5: Connect the points.
Draw straight lines connecting the points. You will see two straight lines meeting at the origin (0,0), forming a 'V' shape.
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Answer: The graph is a V-shape with its vertex at the origin (0,0), opening upwards.
Why It Matters
Understanding absolute value graphs is crucial in fields like Data Science and AI, where you often need to measure 'distance' or 'error' regardless of direction. Engineers use it to design structures that can handle forces from any side, and in Computer Science, it helps in algorithms for image processing or data sorting. It's a foundational concept for many exciting careers!
Common Mistakes
MISTAKE: Assuming |x| is always x, so graphing y = x. | CORRECTION: Remember that for negative x values, |x| changes the sign. For example, if x = -2, y = |-2| = 2, not -2. The graph never goes below the x-axis.
MISTAKE: Drawing a U-shaped curve like a parabola for y = |x|. | CORRECTION: The graph of y = |x| consists of two straight lines. It's not curved like y = x^2. The 'V' shape is made of sharp, straight edges.
MISTAKE: Not extending the lines indefinitely. | CORRECTION: The graph of y = |x| continues infinitely in both positive and negative x directions, and upwards in the y direction. Always draw arrows at the ends of the lines to show they extend.
Practice Questions
Try It Yourself
QUESTION: What is the y-value when x = -5 for the graph y = |x|? | ANSWER: y = 5
QUESTION: Does the point (-4, 4) lie on the graph of y = |x|? Explain why. | ANSWER: Yes, because y = |-4| = 4, so the point (-4, 4) satisfies the equation.
QUESTION: If the graph of y = |x| is shifted 2 units upwards, what would be the new equation and where would its vertex be? | ANSWER: New equation: y = |x| + 2. Vertex would be at (0, 2).
MCQ
Quick Quiz
Which of the following describes the shape of the graph of y = |x|?
A straight line passing through the origin
A V-shaped graph opening upwards
A U-shaped curve (parabola)
A horizontal line
The Correct Answer Is:
B
The graph of y = |x| always produces non-negative y-values and consists of two straight lines meeting at the origin, forming a distinct V-shape. Options A, C, and D describe different types of graphs.
Real World Connection
In the Real World
In navigation apps like Google Maps or Ola, when you search for the 'distance' to a destination, it always gives a positive value, regardless of the direction. This is a real-world application of the absolute value concept. Even if you're 'behind' your starting point in a straight line, the distance covered is always positive, just like y = |x|.
Key Vocabulary
Key Terms
Absolute Value: The distance of a number from zero, always positive or zero. | Origin: The point (0,0) where the x-axis and y-axis intersect. | Vertex: The turning point of a graph; for y = |x|, it's the tip of the 'V'. | Symmetrical: A shape that is identical on both sides of a central line.
What's Next
What to Learn Next
Great job understanding y = |x|! Next, you can explore transformations of this graph, like y = |x| + c or y = |x - h|. This will help you see how changing numbers in the equation shifts or moves the V-shape, building on your current knowledge.
