The Zeros Of A Parabola Are -4 And 2, And { (6, 10)$}$ Is A Point On The Graph. What Is The Value Of { A$}$ In The Equation Of The Parabola?A. ${ 6 = A(10-4)(10+2)\$} B. ${ 6 = A(10+4)(10-2)\$} C. [$10 =

Introduction

In mathematics, a parabola is a quadratic equation that can be represented in the form of $y = ax^2 + bx + c$. The zeros of a parabola are the points where the graph intersects the x-axis, and they can be used to determine the equation of the parabola. In this article, we will explore how to find the value of $a$ in the equation of a parabola given its zeros and a point on the graph.

Understanding the Problem

We are given that the zeros of the parabola are -4 and 2, and the point $(6, 10)$ is on the graph. This means that the parabola passes through the points $(-4, 0)$ and $(2, 0)$, and it also passes through the point $(6, 10)$. Our goal is to find the value of $a$ in the equation of the parabola.

The General Form of a Parabola

The general form of a parabola is given by the equation $y = ax^2 + bx + c$. Since we know that the zeros of the parabola are -4 and 2, we can write the equation as $y = a(x + 4)(x - 2)$. This is because the zeros of the parabola are the roots of the equation, and they can be factored into the form $(x - r_1)(x - r_2)$, where $r_1$ and $r_2$ are the roots.

Using the Point-Slope Form

We are given that the point $(6, 10)$ is on the graph of the parabola. This means that we can substitute the values of $x$ and $y$ into the equation to get an equation in terms of $a$. Substituting $x = 6$ and $y = 10$ into the equation $y = a(x + 4)(x - 2)$, we get:

$$10 = a(6 + 4)(6 - 2)$$

Simplifying the equation, we get:

$$10 = a(10)(4)$$

$$10 = 40a$$

Solving for a

To solve for $a$, we can divide both sides of the equation by 40:

$$a = \frac{10}{40}$$

$$a = \frac{1}{4}$$

Therefore, the value of $a$ in the equation of the parabola is $\frac{1}{4}$.

Conclusion

In this article, we explored how to find the value of $a$ in the equation of a parabola given its zeros and a point on the graph. We used the general form of a parabola and the point-slope form to derive an equation in terms of $a$, and then solved for $a$ by dividing both sides of the equation by 40. The value of $a$ is $\frac{1}{4}$.

Discussion

• What are the zeros of a parabola?
• How can you use the zeros of a parabola to determine the equation of the parabola?
• What is the general form of a parabola?
• How can you use the point-slope form to derive an equation in terms of $a$?
• What is the value of $a$ in the equation of the parabola?

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Calculus" by James Stewart
• [3] "Mathematics for the Nonmathematician" by Morris Kline
  The Zeros of a Parabola: Q&A =============================

Q: What are the zeros of a parabola?

A: The zeros of a parabola are the points where the graph intersects the x-axis. In other words, they are the values of x that make the equation of the parabola equal to zero.

Q: How can you use the zeros of a parabola to determine the equation of the parabola?

A: You can use the zeros of a parabola to determine the equation of the parabola by factoring the equation into the form $(x - r_1)(x - r_2)$, where $r_1$ and $r_2$ are the zeros of the parabola.

Q: What is the general form of a parabola?

A: The general form of a parabola is given by the equation $y = ax^2 + bx + c$. This equation can be factored into the form $(x - r_1)(x - r_2)$, where $r_1$ and $r_2$ are the zeros of the parabola.

Q: How can you use the point-slope form to derive an equation in terms of $a$?

A: You can use the point-slope form to derive an equation in terms of $a$ by substituting the values of $x$ and $y$ into the equation $y = a(x + 4)(x - 2)$, where $(x, y)$ is a point on the graph of the parabola.

Q: What is the value of $a$ in the equation of the parabola?

A: The value of $a$ in the equation of the parabola is $\frac{1}{4}$.

Q: How can you solve for $a$ in the equation of the parabola?

A: You can solve for $a$ in the equation of the parabola by dividing both sides of the equation by the coefficient of $a$.

Q: What are some common mistakes to avoid when solving for $a$ in the equation of the parabola?

A: Some common mistakes to avoid when solving for $a$ in the equation of the parabola include:

• Not factoring the equation into the form $(x - r_1)(x - r_2)$
• Not substituting the values of $x$ and $y$ into the equation
• Not dividing both sides of the equation by the coefficient of $a$

Q: How can you check your work when solving for $a$ in the equation of the parabola?

A: You can check your work when solving for $a$ in the equation of the parabola by plugging the value of $a$ back into the equation and verifying that it satisfies the equation.

Q: What are some real-world applications of the zeros of a parabola?

A: Some real-world applications of the zeros of a parabola include:

• Modeling the trajectory of a projectile
• Determining the maximum height of a thrown object
• Finding the minimum or maximum value of a quadratic function

Q: How can you use the zeros of a parabola to solve problems in physics and engineering?

A: You can use the zeros of a parabola to solve problems in physics and engineering by applying the principles of quadratic equations to real-world problems.

Q: What are some common problems that involve the zeros of a parabola?

A: Some common problems that involve the zeros of a parabola include:

• Finding the maximum or minimum value of a quadratic function
• Determining the time of flight of a projectile
• Finding the height of a thrown object at a given time

Q: How can you use the zeros of a parabola to solve problems in economics and finance?

A: You can use the zeros of a parabola to solve problems in economics and finance by applying the principles of quadratic equations to real-world problems, such as modeling the behavior of a company's stock price or determining the optimal price of a product.