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

Q: How do I know if a point is maximum or minimum?

A maximum point is the highest point on the parabola (like the peak of a mountain), while a minimum point is the lowest point (like the bottom of a valley). The problem tells you which type it is!

Q: Why does a maximum point mean the parabola opens downward?

Think of it visually: if $ (4,0) $ is the highest point, the parabola must curve downward from there. An upward-opening parabola would have its lowest point at the vertex, making it a minimum instead.

Q: What if the vertex has a different y-coordinate?

Use the same process! For vertex $ (4,3) $, the equation would be $ y = -(x-4)^2 + 3 $. The k-value shifts the parabola up or down.

Q: How do I remember the signs in vertex form?

Remember: $ y = a(x - h)^2 + k $ uses opposite signs. For vertex (4,0), write $ (x - 4) $, not $ (x + 4) $. The h-value gets subtracted inside the parentheses.

Q: Can I check my answer by plugging in the vertex coordinates?

Absolutely! Substitute $ x = 4 $ into $ y = -(x-4)^2 $: you get $ y = -(0)^2 = 0 $. This confirms the vertex is indeed $ (4,0) $!

Vertex Form Identification with Maximum Points

Which function corresponds to the parabola with a maximum point of
$(4,0)$ ?

❤️ 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

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

00:03 A smiling parabola has a positive coefficient for X squared

00:07 A sad parabola has a negative coefficient for X squared

00:12 In this parabola, the intersection point is at the origin

00:19 We need a sad parabola 4 steps to the right

00:27 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 function corresponds to the parabola with a maximum point of
$(4,0)$ ?

Step-by-step solution

To solve this problem, we'll use the vertex form of a parabolic function.

Recall that the vertex form of a parabola is given by:

$y = a(x - h)^2 + k$

where $(h, k)$ is the vertex of the parabola. In this problem, we are given a vertex at $(4, 0)$ .

Step 1: Identify the vertex coordinates:

$h = 4$
$k = 0$

Step 2: Determine the sign of $a$ .

We are informed that the point $(4, 0)$ is a maximum point, which means the parabola opens downward. For a downward-opening parabola, the coefficient $a$ must be negative.

Step 3: Substitute the identified values into the vertex form equation:

$y = a(x - 4)^2 + 0$

$y = a(x - 4)^2$

Since the parabola is downward-opening:

$a < 0$ , for instance, $a = -1$

Thus, the equation is $y = -(x - 4)^2$ .

This equation describes a parabola with a vertex at $(4, 0)$ and opens downward, achieving a maximum there. Therefore, the correct function corresponding to the parabola with a maximum point at $(4, 0)$ is:

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

Final Answer

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

Key Points to Remember

Essential concepts to master this topic

Vertex Form: $y = a(x - h)^2 + k$ where (h,k) is the vertex
Direction Rule: Maximum point means a < 0, so parabola opens downward
Verification: Substitute x = 4: $y = -(4-4)^2 = 0$ confirms maximum ✓

Common Mistakes

Avoid these frequent errors

Confusing maximum and minimum with parabola direction
Don't assume a maximum point means a > 0 = upward opening parabola! This gives the opposite direction. A maximum point means the parabola reaches its highest value there, so it must open downward with a < 0. Always remember: maximum = downward opening (a < 0), minimum = upward opening (a > 0).

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

How do I know if a point is maximum or minimum?

A maximum point is the highest point on the parabola (like the peak of a mountain), while a minimum point is the lowest point (like the bottom of a valley). The problem tells you which type it is!

Why does a maximum point mean the parabola opens downward?

Think of it visually: if $(4,0)$ is the highest point, the parabola must curve downward from there. An upward-opening parabola would have its lowest point at the vertex, making it a minimum instead.

What if the vertex has a different y-coordinate?

Use the same process! For vertex $(4,3)$ , the equation would be $y = -(x-4)^2 + 3$ . The k-value shifts the parabola up or down.

How do I remember the signs in vertex form?

Remember: $y = a(x - h)^2 + k$ uses opposite signs. For vertex (4,0), write $(x - 4)$ , not $(x + 4)$ . The h-value gets subtracted inside the parentheses.

Can I check my answer by plugging in the vertex coordinates?

Absolutely! Substitute $x = 4$ into $y = -(x-4)^2$ : you get $y = -(0)^2 = 0$ . This confirms the vertex is indeed $(4,0)$ !

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Parabola Families 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