Identify the Quadratic Function with Minimum Point (2,0): Parabola Analysis

Q: Why is the vertex form $ y = a(x-h)^2 + k $ and not $ y = a(x+h)^2 + k $ ?

The minus sign in vertex form is standard because it makes the algebra work correctly. When the vertex is at x = 2, we need $ (x-2)^2 $ to equal zero when x = 2.

Q: How do I know if the parabola opens up or down?

Look at the coefficient a in $ y = a(x-h)^2 + k $. If a > 0, the parabola opens upward (has a minimum). If a downward (has a maximum).

Q: What if the problem asks for a maximum instead of minimum?

For a maximum point, you need a negative value for coefficient a, like $ y = -(x-2)^2 $. This flips the parabola upside down.

Q: Can I expand the vertex form to check my answer?

Absolutely! Expanding $ y = (x-2)^2 $ gives $ y = x^2 - 4x + 4 $. You can verify the vertex by completing the square or using $ x = -\frac{b}{2a} $.

Q: What happens if k ≠ 0 in the vertex?

If the vertex is at (2,3) instead of (2,0), then k = 3 and your function becomes $ y = (x-2)^2 + 3 $. The k-value shifts the parabola up or down.

Vertex Form with Minimum Coordinates

Which of the follling represents a function that has a parabola with a minimum point of $(2,0)$ ?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the appropriate function for the parabola with the minimum point

00:06 Let's check this function

00:09 Let's substitute the X of the minimum point and check Y

00:17 Y = 0 as we wanted

00:24 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

Which of the follling represents a function that has a parabola with a minimum point of $(2,0)$ ?

Step-by-step solution

To solve this problem, we need to ensure that the parabola's vertex is located at $(2,0)$ . We use the vertex form of a quadratic equation $y = a(x-h)^2 + k$ , where $(h, k)$ is the vertex of the parabola.

Given the minimum point or vertex as $(2,0)$ , we identify $h = 2$ and $k = 0$ . Substituting these into the vertex form gives:

$y = a(x-2)^2 + 0$

Since we are looking for a parabola with a minimum point, $a$ should be positive. The simplest positive value for $a$ is 1, giving:

$y = (x-2)^2$

This matches choice 1. Therefore, the correct representation of the function is:

$y=(x-2)^2$

Final Answer

$y=(x-2)^2$

Key Points to Remember

Essential concepts to master this topic

Vertex Form: Use $y = a(x-h)^2 + k$ where vertex is (h,k)
Technique: For minimum at (2,0), substitute h=2 and k=0
Check: Verify vertex by finding where derivative equals zero ✓

Common Mistakes

Avoid these frequent errors

Confusing signs in vertex form
Don't write $y = (x+2)^2$ for vertex (2,0) = vertex at (-2,0)! The sign in (x-h) is opposite to the x-coordinate. Always use $y = (x-2)^2$ for vertex at x=2.

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 is the vertex form $y = a(x-h)^2 + k$ and not $y = a(x+h)^2 + k$ ?

The minus sign in vertex form is standard because it makes the algebra work correctly. When the vertex is at x = 2, we need $(x-2)^2$ to equal zero when x = 2.

How do I know if the parabola opens up or down?

Look at the coefficient a in $y = a(x-h)^2 + k$ . If a > 0, the parabola opens upward (has a minimum). If a < 0, it opens downward (has a maximum).

What if the problem asks for a maximum instead of minimum?

For a maximum point, you need a negative value for coefficient a, like $y = -(x-2)^2$ . This flips the parabola upside down.

Can I expand the vertex form to check my answer?

Absolutely! Expanding $y = (x-2)^2$ gives $y = x^2 - 4x + 4$ . You can verify the vertex by completing the square or using $x = -\frac{b}{2a}$ .

What happens if k ≠ 0 in the vertex?

If the vertex is at (2,3) instead of (2,0), then k = 3 and your function becomes $y = (x-2)^2 + 3$ . The k-value shifts the parabola up or down.

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Identify the Quadratic Function with Minimum Point (2,0): Parabola Analysis

Vertex Form with Minimum Coordinates

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why is the vertex form $y = a(x-h)^2 + k$ and not $y = a(x+h)^2 + k$ ?

How do I know if the parabola opens up or down?

What if the problem asks for a maximum instead of minimum?

Can I expand the vertex form to check my answer?

What happens if k ≠ 0 in the vertex?

🌟 Unlock Your Math Potential

More Questions

Parabola of the Form y=(x-p)²

Identify the Quadratic Function: Parabola with Maximum Point (4,0)

Find the Quadratic Function with Minimum Point (-2,0): Parabola Identification

Finding Y-Intercept: Substituting x=0 into Equations

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

Identify the Quadratic Function with Minimum Point (2,0): Parabola Analysis

Vertex Form with Minimum Coordinates

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why is the vertex form y=a(x−h)2+k y = a(x-h)^2 + k y=a(x−h)2+k and not y=a(x+h)2+k y = a(x+h)^2 + k y=a(x+h)2+k?

How do I know if the parabola opens up or down?

What if the problem asks for a maximum instead of minimum?

Can I expand the vertex form to check my answer?

What happens if k ≠ 0 in the vertex?

🌟 Unlock Your Math Potential

More Questions

Parabola of the Form y=(x-p)²

Identify the Quadratic Function: Parabola with Maximum Point (4,0)

Find the Quadratic Function with Minimum Point (-2,0): Parabola Identification

Finding Y-Intercept: Substituting x=0 into Equations

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

Why is the vertex form $y = a(x-h)^2 + k$ and not $y = a(x+h)^2 + k$ ?