Comparing Vertices: When is x² Equal to 2x² at the Same Point?

Q: Why doesn't the coefficient 2 change the vertex?

The coefficient only affects the shape of the parabola, not its position! Think of it like stretching a rubber band - the center point stays the same, but the curve gets narrower or wider.

Q: How can I tell which parabola is narrower?

The larger the coefficient, the narrower the parabola! So $ y = 2x^2 $ is narrower than $ y = x^2 $ because 2 > 1.

Q: What if the coefficient was negative, like -2?

A negative coefficient would flip the parabola upside down (opening downward), but the vertex would still be at $ (0, 0) $!

Q: Do all parabolas of the form y = ax² have the same vertex?

Yes! Any function in the form $ y = ax^2 $ (where a ≠ 0) has its vertex at the origin $ (0, 0) $.

Q: How do I find the vertex if there are other terms?

If you have terms like $ y = x^2 + 3x + 2 $, you'll need to complete the square or use the vertex formula. But for simple $ y = ax^2 $, it's always $ (0, 0) $!

Quadratic Functions with Same Vertices

The vertex of the function $y=x^2$ is the same vertex of the function $y=2x^2$ .

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Do the functions have the same vertex point?

00:06 We'll use the formula to calculate the vertex point of a parabola

00:12 Let's look at the function coefficients

00:15 We'll substitute in the formula and calculate the vertex point

00:23 We'll substitute the X value we found to find the vertex point

00:27 This is the vertex point of the first function

00:31 We'll use the same method to find the vertex point for the second function

00:52 We can see that the vertex points are equal

00:56 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

The vertex of the function $y=x^2$ is the same vertex of the function $y=2x^2$ .

Step-by-step solution

The given functions are $y = x^2$ and $y = 2x^2$ . Both functions are written in the form $y = ax^2$ , where the vertex for such parabolas is determined by setting $x = 0$ .

Let's analyze each function step-by-step:

Function $y = x^2$ :
This function represents a parabola that opens upwards with its vertex located at $(0, 0)$ .
Function $y = 2x^2$ :
Similarly, this function is also a parabola that opens upwards. The coefficient $2$ affects the "narrowness" or "width" of the parabola but does not shift the vertex, so the vertex remains at $(0, 0)$ .

Both functions $y = x^2$ and $y = 2x^2$ share the same vertex at the point $(0, 0)$ . Therefore, the statement that the vertex of the function $y=x^2$ is the same vertex of the function $y=2x^2$ is True.

Thus, the correct answer to the problem is True.

Final Answer

True.

Key Points to Remember

Essential concepts to master this topic

Vertex Rule: For $y = ax^2$ , vertex is always at (0, 0)
Technique: Coefficient a changes width, not vertex position: $y = x^2$ and $y = 2x^2$
Check: Substitute x = 0 into both functions: 0² = 0 and 2(0)² = 0 ✓

Common Mistakes

Avoid these frequent errors

Confusing coefficient effects on vertex location
Don't assume the coefficient 2 moves the vertex = wrong vertex coordinates! The coefficient only affects how narrow or wide the parabola appears, not where it touches the axis. Always remember that $y = ax^2$ has vertex at (0, 0) regardless of the value of a.

Practice Quiz

Test your knowledge with interactive questions

Find the intersection of the function

$$ y=(x+4)^2 $$

With the Y

FAQ

Everything you need to know about this question

Why doesn't the coefficient 2 change the vertex?

The coefficient only affects the shape of the parabola, not its position! Think of it like stretching a rubber band - the center point stays the same, but the curve gets narrower or wider.

How can I tell which parabola is narrower?

The larger the coefficient, the narrower the parabola! So $y = 2x^2$ is narrower than $y = x^2$ because 2 > 1.

What if the coefficient was negative, like -2?

A negative coefficient would flip the parabola upside down (opening downward), but the vertex would still be at $(0, 0)$ !

Do all parabolas of the form y = ax² have the same vertex?

Yes! Any function in the form $y = ax^2$ (where a ≠ 0) has its vertex at the origin $(0, 0)$ .

How do I find the vertex if there are other terms?

If you have terms like $y = x^2 + 3x + 2$ , you'll need to complete the square or use the vertex formula. But for simple $y = ax^2$ , it's always $(0, 0)$ !

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