Decoding Cartesian Coordinate Graphs: A Visual Understanding

Q: Why does the graph look like a V instead of a straight line?

The absolute value function $ y = |x| $ creates a V-shape because it takes negative x-values and makes them positive. For example, when x = -3, y = |-3| = 3, creating the left side of the V.

Q: How can I tell this isn't just y = x?

The function $ y = x $ would be a straight diagonal line going through points like (-1, -1) and (1, 1). But this graph shows no negative y-values, which is the key feature of absolute value.

Q: What makes this different from y = -x?

The function $ y = -x $ would be a straight line sloping downward from left to right. This V-shaped graph has two different slopes: negative on the left side and positive on the right side.

Q: Where is the vertex and why is it important?

The vertex is at (0, 0) because when x = 0, y = |0| = 0. This is the lowest point of the graph and where the two sides of the V meet.

Q: How do I verify this is y = |x|?

Test key points: When x = 2: y = |2| = 2 ✓When x = -2: y = |-2| = 2 ✓Both give the same y-value, showing the symmetry of absolute value!

Absolute Value Functions with V-Shaped Graphs

The graph corresponds to

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

The graph corresponds to

Step-by-step solution

To solve this problem, we'll analyze the graph to match it with the provided equations.

Step 1: The graph is in a 'V' shape presenting symmetry around the y-axis and has its vertex at the origin.
Step 2: The equation that corresponds to this type of graph is the absolute value function $y = |x|$ . Absolute value functions show symmetry around the y-axis and have a vertex where the argument of the absolute value, here $x$ , equals zero.
Step 3: The graph we see perfectly matches an absolute value function, as it does not extend below the x-axis and has the aforementioned symmetrical properties. Therefore, it is represented by the equation $y = |x|$ .

Therefore, the solution to the problem is $y = |x|$ , matching the graph with the choice corresponding to $y = |x|$ .

Final Answer

$y=|x|$

Key Points to Remember

Essential concepts to master this topic

Shape Recognition: V-shaped graphs opening upward indicate absolute value functions
Technique: Check vertex at origin (0,0) and symmetry across y-axis
Verification: Test points: when x = 3, y = |3| = 3 ✓

Common Mistakes

Avoid these frequent errors

Confusing V-shape with linear functions
Don't assume V-shaped graphs are just two linear pieces = missing the absolute value! Linear functions y = x or y = -x create straight lines, not V-shapes. Always recognize that V-shaped graphs with vertex at origin represent y = |x|.

Practice Quiz

Test your knowledge with interactive questions

$\left|-x\right|=10$

FAQ

Everything you need to know about this question

Why does the graph look like a V instead of a straight line?

The absolute value function $y = |x|$ creates a V-shape because it takes negative x-values and makes them positive. For example, when x = -3, y = |-3| = 3, creating the left side of the V.

How can I tell this isn't just y = x?

The function $y = x$ would be a straight diagonal line going through points like (-1, -1) and (1, 1). But this graph shows no negative y-values, which is the key feature of absolute value.

What makes this different from y = -x?

The function $y = -x$ would be a straight line sloping downward from left to right. This V-shaped graph has two different slopes: negative on the left side and positive on the right side.

Where is the vertex and why is it important?

The vertex is at (0, 0) because when x = 0, y = |0| = 0. This is the lowest point of the graph and where the two sides of the V meet.

How do I verify this is y = |x|?

Test key points:

When x = 2: y = |2| = 2 ✓
When x = -2: y = |-2| = 2 ✓
Both give the same y-value, showing the symmetry of absolute value!

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Absolute Value and Inequality questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime