Graph to Equation Puzzle: Identify the Matching Function

Q: How can I tell if a parabola is y = x² or y = 2x²?

Look at specific points! For $ y = x^2 $, when x = 2, y = 4. For $ y = 2x^2 $, when x = 2, y = 8. The narrower parabola has the larger coefficient.

Q: What if the parabola opens downward instead?

A downward-opening parabola has a negative coefficient! So $ y = -x^2 $ opens down, while $ y = x^2 $ opens up. The graph direction tells you the sign.

Q: Why is the vertex important for identifying the equation?

The vertex shows you the starting point of the parabola! For $ y = x^2 $, it's at (0,0). If the vertex moved, you'd need additional terms like $ y = x^2 + 3 $.

Q: How do I check if my chosen equation is correct?

Pick two clear points from the graph and substitute into your equation. If both points satisfy the equation, you've found the right match!

Q: What's the difference between y = x² and y = 4x²?

$ y = 4x^2 $ is much narrower than $ y = x^2 $. When x = 1, the first gives y = 4, while the second gives y = 1. The larger coefficient makes a steeper curve.

Quadratic Functions with Standard Graph Analysis

Which of the following equations corresponds to the function represented in the graph?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the appropriate equation for the function in the graph

00:03 Find points on the graph

00:56 We'll identify the pattern and deduce the graph's equation from it

01:01 And this is the solution to the question

Step-by-step written solution

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Understand the problem

Which of the following equations corresponds to the function represented in the graph?

Step-by-step solution

To solve this problem, let's examine each key feature of the graph:

Step 1: Vertex and Direction - The graph appears to have its vertex at the origin and opens upwards.
Step 2: Width - For a quadratic $y = ax^2$ , if $|a| = 1$ , the width is standard like $y = x^2$ . In this graph, the parabola seems to follow the standard width.

Comparing to the options, we analyze for matching attributes:

Choice 1: $y = 4x^2$ suggests a narrow upward parabola.
Choice 2: $y = 2x^2$ is narrower than a standard parabola.
Choice 3: $y = x^2$ matches a standard parabola.
Choice 4: $y = -x^2$ opens downward.

Given the graph opens upwards and matches the standard parabola, the correct equation is $y = x^2$ .

Final Answer

$y=x^2$

Key Points to Remember

Essential concepts to master this topic

Vertex Position: Standard parabola $y = x^2$ has vertex at origin (0,0)
Graph Features: Upward opening means positive coefficient, standard width indicates coefficient = 1
Verification: Check key points like (1,1) and (-1,1) match the graph ✓

Common Mistakes

Avoid these frequent errors

Confusing parabola width with coefficient value
Don't assume a wider parabola always means smaller coefficient = wrong equation choice! The graph shows standard width, not narrow or wide. Always compare the parabola's width to the standard $y = x^2$ shape to identify the correct coefficient.

Practice Quiz

Test your knowledge with interactive questions

Determine whether the given graph is a function?

FAQ

Everything you need to know about this question

How can I tell if a parabola is y = x² or y = 2x²?

Look at specific points! For $y = x^2$ , when x = 2, y = 4. For $y = 2x^2$ , when x = 2, y = 8. The narrower parabola has the larger coefficient.

What if the parabola opens downward instead?

A downward-opening parabola has a negative coefficient! So $y = -x^2$ opens down, while $y = x^2$ opens up. The graph direction tells you the sign.

Why is the vertex important for identifying the equation?

The vertex shows you the starting point of the parabola! For $y = x^2$ , it's at (0,0). If the vertex moved, you'd need additional terms like $y = x^2 + 3$ .

How do I check if my chosen equation is correct?

Pick two clear points from the graph and substitute into your equation. If both points satisfy the equation, you've found the right match!

What's the difference between y = x² and y = 4x²?

$y = 4x^2$ is much narrower than $y = x^2$ . When x = 1, the first gives y = 4, while the second gives y = 1. The larger coefficient makes a steeper curve.

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