Using the Zeros to Find the Equation

Oct 13, 2025 by magicalceylontours.com 59 views

Understanding the Basics of a Parabola

A parabola is a quadratic equation in the form of $y = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants. The zeros of a parabola are the values of $x$ where the parabola intersects the x-axis, and the y-intercept is the value of $y$ where the parabola intersects the y-axis.

Given Information

We are given that the parabola has zeros of $x = -1$ and $x = 3$ , and a $y$ -intercept of $(0, -9)$ . This means that the parabola intersects the x-axis at $x = -1$ and $x = 3$ , and it intersects the y-axis at $y = -9$ .

Using the Zeros to Find the Equation

Since the zeros of the parabola are $x = -1$ and $x = 3$ , we can write the equation of the parabola in factored form as $y = a(x + 1)(x - 3)$ , where $a$ is a constant.

Using the Y-Intercept to Find the Value of a

We are given that the y-intercept of the parabola is $(0, -9)$ . Substituting $x = 0$ and $y = -9$ into the equation $y = a(x + 1)(x - 3)$ , we get:

-9 = a(0 + 1)(0 - 3)

Simplifying the equation, we get:

-9 = -3a

Dividing both sides by $-3$ , we get:

a = 3

Finding the Equation of the Parabola

Now that we have found the value of $a$ , we can substitute it into the equation $y = a(x + 1)(x - 3)$ to get the equation of the parabola:

y = 3(x + 1)(x - 3)

Expanding the equation, we get:

y = 3(x^2 - 2x - 3)

Simplifying the equation, we get:

y = 3x^2 - 6x - 9

Conclusion

Therefore, the equation of the parabola that has zeros of $x = -1$ and $x = 3$ and a $y$ -intercept of $(0, -9)$ is $y = 3x^2 - 6x - 9$ .

Comparing with the Given Options

Comparing the equation we found with the given options, we see that the correct answer is:

A) $y = 3x^2 - 6x - 9$

This is the equation we found using the zeros and y-intercept of the parabola.

Final Answer

The final answer is A) $y = 3x^2 - 6x - 9$ .

Understanding the Basics of a Parabola

Q&A

Q: What is the general form of a parabola equation?

A: The general form of a parabola equation is $y = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants.

Q: What are the zeros of a parabola?

A: The zeros of a parabola are the values of $x$ where the parabola intersects the x-axis.

Q: What is the y-intercept of a parabola?

A: The y-intercept of a parabola is the value of $y$ where the parabola intersects the y-axis.

Q: How do you find the equation of a parabola with given zeros and y-intercept?

A: To find the equation of a parabola with given zeros and y-intercept, you can use the factored form of the equation $y = a(x - r_1)(x - r_2)$ , where $r_1$ and $r_2$ are the zeros of the parabola. Then, substitute the y-intercept into the equation to find the value of $a$ .

Q: What is the equation of a parabola with zeros of $x = -1$ and $x = 3$ and a $y$ -intercept of $(0, -9)$ ?

A: The equation of a parabola with zeros of $x = -1$ and $x = 3$ and a $y$ -intercept of $(0, -9)$ is $y = 3x^2 - 6x - 9$ .

Q: How do you compare the equation of a parabola with the given options?

A: To compare the equation of a parabola with the given options, you can substitute the values of $x$ and $y$ into the equation and check if it satisfies the given conditions.

Conclusion

In this article, we have discussed the basics of a parabola, including its general form, zeros, and y-intercept. We have also provided a step-by-step guide on how to find the equation of a parabola with given zeros and y-intercept. Additionally, we have answered some frequently asked questions about parabolas.

Final Tips

Make sure to understand the basics of a parabola before trying to find its equation.
Use the factored form of the equation to find the equation of a parabola with given zeros.
Substitute the y-intercept into the equation to find the value of $a$ .
Compare the equation of a parabola with the given options to ensure that it satisfies the given conditions.

Common Mistakes to Avoid

Not understanding the basics of a parabola.
Not using the factored form of the equation to find the equation of a parabola with given zeros.
Not substituting the y-intercept into the equation to find the value of $a$ .
Not comparing the equation of a parabola with the given options to ensure that it satisfies the given conditions.

Final Answer

The final answer is A) $y = 3x^2 - 6x - 9$ .