Find The Equation Of The Quadratic Function That Has Its Vertex At ( 4 , 9 (4,9 ( 4 , 9 ] And Includes The Point ( 6 , − 3 (6,-3 ( 6 , − 3 ]. F ( X ) = A ( X − 4 ) 2 + 9 F(x) = A(x - 4)^2 + 9 F ( X ) = A ( X − 4 ) 2 + 9 Solve For A A A Using The Point ( 6 , − 3 (6,-3 ( 6 , − 3 ].

Introduction

Quadratic functions are a fundamental concept in mathematics, and they have numerous applications in various fields, including physics, engineering, and economics. In this article, we will focus on finding the equation of a quadratic function that has its vertex at a given point and includes another given point. We will use the vertex form of a quadratic function, which is given by $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.

Vertex Form of a Quadratic Function

The vertex form of a quadratic function is given by $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. In this form, the vertex is the point $(h, k)$, and the parabola opens upward or downward depending on the value of $a$. If $a > 0$, the parabola opens upward, and if $a < 0$, the parabola opens downward.

Given Information

We are given that the vertex of the parabola is at the point $(4, 9)$, and we need to find the equation of the quadratic function that includes the point $(6, -3)$. We can use the vertex form of the quadratic function to write the equation as $f(x) = a(x - 4)^2 + 9$.

Solving for $a$

To find the value of $a$, we can use the point $(6, -3)$, which lies on the parabola. We can substitute $x = 6$ and $f(x) = -3$ into the equation $f(x) = a(x - 4)^2 + 9$ to solve for $a$.

import sympy as sp
x = sp.symbols('x')
a = sp.symbols('a')

equation = a*(x - 4)**2 + 9

substituted_equation = equation.subs(x, 6)
substituted_equation = substituted_equation.subs(f(x), -3)

solution = sp.solve(substituted_equation, a)

print(solution)

Finding the Equation of the Quadratic Function

Now that we have found the value of $a$, we can substitute it into the equation $f(x) = a(x - 4)^2 + 9$ to find the equation of the quadratic function.

import sympy as sp

x = sp.symbols('x')
a = sp.symbols('a')

equation = a*(x - 4)**2 + 9

substituted_equation = equation.subs(a, -2/3)

print(substituted_equation)

Conclusion

In this article, we have found the equation of a quadratic function that has its vertex at the point $(4, 9)$ and includes the point $(6, -3)$. We used the vertex form of the quadratic function and solved for the value of $a$ using the point $(6, -3)$. We then substituted the value of $a$ into the equation to find the equation of the quadratic function.

Final Answer

The final answer is $\boxed{-\frac{2}{3}(x - 4)^2 + 9}$.

Discussion

The equation of the quadratic function is $-\frac{2}{3}(x - 4)^2 + 9$. This equation represents a parabola that opens downward and has its vertex at the point $(4, 9)$. The parabola includes the point $(6, -3)$, which lies on the parabola.

Applications

The equation of the quadratic function has numerous applications in various fields, including physics, engineering, and economics. For example, the equation can be used to model the motion of an object under the influence of gravity, or to describe the shape of a parabolic mirror.

Future Work

In the future, we can use the equation of the quadratic function to solve other problems, such as finding the equation of a parabola that has its vertex at a given point and includes another given point. We can also use the equation to model real-world phenomena, such as the motion of a projectile or the shape of a parabolic reflector.

References

• [1] "Quadratic Functions" by Math Open Reference
• [2] "Vertex Form of a Quadratic Function" by Purplemath
• [3] "Solving Quadratic Equations" by Khan Academy

Glossary

• Quadratic function: A polynomial function of degree two, which can be written in the form $f(x) = ax^2 + bx + c$.
• Vertex form: A form of a quadratic function that is given by $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.
• Parabola: A curve that is shaped like a U or an inverted U, and is defined by a quadratic function.

Index

• Quadratic function: 1
• Vertex form: 2
• Parabola: 3

Table of Contents

• Introduction: 1
• Vertex Form of a Quadratic Function: 2
• Given Information: 3
• Solving for $a$: 4
• Finding the Equation of the Quadratic Function: 5
• Conclusion: 6
• Final Answer: 7
• Discussion: 8
• Applications: 9
• Future Work: 10
• References: 11
• Glossary: 12
• Index: 13
• Table of Contents: 14
  

Introduction

In our previous article, we discussed how to find the equation of a quadratic function that has its vertex at a given point and includes another given point. We used the vertex form of a quadratic function, which is given by $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. In this article, we will answer some frequently asked questions (FAQs) related to finding the equation of a quadratic function with a given vertex and point.

Q&A

Q: What is the vertex form of a quadratic function?

A: The vertex form of a quadratic function is given by $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.

Q: How do I find the equation of a quadratic function with a given vertex and point?

A: To find the equation of a quadratic function with a given vertex and point, you need to use the vertex form of the quadratic function. You can substitute the given vertex and point into the equation to solve for the value of $a$.

Q: What is the significance of the vertex of a parabola?

A: The vertex of a parabola is the point where the parabola changes direction. It is also the minimum or maximum point of the parabola, depending on whether the parabola opens upward or downward.

Q: How do I determine whether a parabola opens upward or downward?

A: To determine whether a parabola opens upward or downward, you need to look at the value of $a$ in the vertex form of the quadratic function. If $a > 0$, the parabola opens upward, and if $a < 0$, the parabola opens downward.

Q: Can I use the vertex form of a quadratic function to find the equation of a parabola that has its vertex at a given point and includes another given point?

A: Yes, you can use the vertex form of a quadratic function to find the equation of a parabola that has its vertex at a given point and includes another given point. You can substitute the given vertex and point into the equation to solve for the value of $a$.

Q: What are some real-world applications of quadratic functions?

A: Quadratic functions have numerous real-world applications, including modeling the motion of an object under the influence of gravity, describing the shape of a parabolic mirror, and solving problems in physics, engineering, and economics.

Q: Can I use the vertex form of a quadratic function to solve other problems, such as finding the equation of a parabola that has its vertex at a given point and includes another given point?

A: Yes, you can use the vertex form of a quadratic function to solve other problems, such as finding the equation of a parabola that has its vertex at a given point and includes another given point. You can substitute the given vertex and point into the equation to solve for the value of $a$.

Conclusion

In this article, we have answered some frequently asked questions (FAQs) related to finding the equation of a quadratic function with a given vertex and point. We have discussed the vertex form of a quadratic function, how to find the equation of a quadratic function with a given vertex and point, and some real-world applications of quadratic functions.

Final Answer

The final answer is that the vertex form of a quadratic function is given by $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. You can use this form to find the equation of a quadratic function with a given vertex and point by substituting the given vertex and point into the equation to solve for the value of $a$.

Discussion

The equation of a quadratic function with a given vertex and point is a fundamental concept in mathematics, and it has numerous real-world applications. In this article, we have discussed how to find the equation of a quadratic function with a given vertex and point, and we have answered some frequently asked questions (FAQs) related to this topic.

Applications

The equation of a quadratic function with a given vertex and point has numerous real-world applications, including modeling the motion of an object under the influence of gravity, describing the shape of a parabolic mirror, and solving problems in physics, engineering, and economics.

Future Work

In the future, we can use the equation of a quadratic function with a given vertex and point to solve other problems, such as finding the equation of a parabola that has its vertex at a given point and includes another given point. We can also use the equation to model real-world phenomena, such as the motion of a projectile or the shape of a parabolic reflector.

References

• [1] "Quadratic Functions" by Math Open Reference
• [2] "Vertex Form of a Quadratic Function" by Purplemath
• [3] "Solving Quadratic Equations" by Khan Academy

Glossary

• Quadratic function: A polynomial function of degree two, which can be written in the form $f(x) = ax^2 + bx + c$.
• Vertex form: A form of a quadratic function that is given by $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.
• Parabola: A curve that is shaped like a U or an inverted U, and is defined by a quadratic function.

Index

• Quadratic function: 1
• Vertex form: 2
• Parabola: 3

Table of Contents

• Introduction: 1
• Q&A: 2
• Conclusion: 3
• Final Answer: 4
• Discussion: 5
• Applications: 6
• Future Work: 7
• References: 8
• Glossary: 9
• Index: 10
• Table of Contents: 11