Find The Equation Of The Parabola That Has Zeros Of X = − 2 X = -2 X = − 2 And X = 3 X = 3 X = 3 And A Y Y Y -intercept Of ( 0 , − 30 (0, -30 ( 0 , − 30 ].A) Y = 5 X 2 − 5 X − 30 Y = 5x^2 - 5x - 30 Y = 5 X 2 − 5 X − 30 B) Y = − 5 X 2 + 5 X + 30 Y = -5x^2 + 5x + 30 Y = − 5 X 2 + 5 X + 30 C) Y = 5 X 2 + 5 X − 30 Y = 5x^2 + 5x - 30 Y = 5 X 2 + 5 X − 30 D) $y =

Understanding the Basics of a Parabola

A parabola is a type of 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 the y-intercept is the point where the graph intersects the y-axis. In this article, we will focus on finding the equation of a parabola that has zeros of $x = -2$ and $x = 3$ and a $y$-intercept of $(0, -30)$.

The Importance of Zeros and Y-Intercept in a Parabola

The zeros of a parabola are crucial in determining its equation. Since we are given the zeros of the parabola as $x = -2$ and $x = 3$, we can use this information to find the equation of the parabola. Additionally, the y-intercept of the parabola is also given as $(0, -30)$, which will help us determine the value of $c$ in the equation $y = ax^2 + bx + c$.

Using the Zeros to Find the Equation of the Parabola

Since the zeros of the parabola are $x = -2$ and $x = 3$, we can write the equation of the parabola in factored form as $y = a(x + 2)(x - 3)$. To find the value of $a$, we can use the y-intercept of the parabola, which is $(0, -30)$. Substituting $x = 0$ and $y = -30$ into the equation, we get:

$$-30 = a(0 + 2)(0 - 3)$$

Simplifying the equation, we get:

$$-30 = a(2)(-3)$$

$$-30 = -6a$$

Dividing both sides by $-6$, we get:

$$a = 5$$

Finding the Equation of the Parabola

Now that we have found the value of $a$, we can substitute it into the equation $y = a(x + 2)(x - 3)$ to get the equation of the parabola:

$$y = 5(x + 2)(x - 3)$$

Expanding the equation, we get:

$$y = 5(x^2 - x - 6)$$

$$y = 5x^2 - 5x - 30$$

Conclusion

In this article, we have found the equation of a parabola that has zeros of $x = -2$ and $x = 3$ and a $y$-intercept of $(0, -30)$. The equation of the parabola is $y = 5x^2 - 5x - 30$. This equation can be used to graph the parabola and find its x-intercepts, y-intercept, and other important features.

Answer

The correct answer is A) $y = 5x^2 - 5x - 30$.

Discussion

This problem requires the use of algebraic techniques to find the equation of a parabola. The zeros of the parabola are used to determine the equation in factored form, and the y-intercept is used to find the value of $a$. The equation of the parabola is then found by expanding the factored form. This problem is a good example of how algebraic techniques can be used to solve problems in mathematics.

Related Topics

• Quadratic Equations: Quadratic equations are a type of polynomial equation that can be represented in the form of $ax^2 + bx + c = 0$. The zeros of a quadratic equation are the points where the graph intersects the x-axis.
• Graphing Quadratic Equations: Quadratic equations can be graphed using various techniques, including factoring, the quadratic formula, and graphing calculators.
• Solving Systems of Equations: Systems of equations are a set of two or more equations that are solved simultaneously. Quadratic equations can be used to solve systems of equations.

Practice Problems

• Find the equation of a parabola that has zeros of $x = -1$ and $x = 4$ and a $y$-intercept of $(0, -20)$.
• Find the equation of a parabola that has zeros of $x = -3$ and $x = 2$ and a $y$-intercept of $(0, -15)$.
• Find the equation of a parabola that has zeros of $x = -5$ and $x = 1$ and a $y$-intercept of $(0, -25)$.

Conclusion

In this article, we have found the equation of a parabola that has zeros of $x = -2$ and $x = 3$ and a $y$-intercept of $(0, -30)$. The equation of the parabola is $y = 5x^2 - 5x - 30$. This equation can be used to graph the parabola and find its x-intercepts, y-intercept, and other important features.

Q: What is a parabola?

A: A parabola is a type of 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 the y-intercept is the point where the graph intersects the y-axis.

Q: How do I 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 following steps:

1. Write the equation of the parabola in factored form as $y = a(x - r_1)(x - r_2)$, where $r_1$ and $r_2$ are the zeros of the parabola.
2. Use the y-intercept to find the value of $a$.
3. Expand the factored form to get the equation of the parabola in standard form.

Q: What is the importance of zeros and y-intercept in a parabola?

A: The zeros of a parabola are crucial in determining its equation. The y-intercept is also important as it helps to find the value of $a$ in the equation $y = ax^2 + bx + c$.

Q: How do I find the value of $a$ in the equation $y = ax^2 + bx + c$?

A: To find the value of $a$ in the equation $y = ax^2 + bx + c$, you can use the y-intercept of the parabola. Substitute $x = 0$ and $y = c$ into the equation, and solve for $a$.

Q: What is the equation of a parabola with zeros of $x = -2$ and $x = 3$ and a $y$-intercept of $(0, -30)$?

A: The equation of a parabola with zeros of $x = -2$ and $x = 3$ and a $y$-intercept of $(0, -30)$ is $y = 5x^2 - 5x - 30$.

Q: How do I graph a parabola?

A: To graph a parabola, you can use various techniques, including factoring, the quadratic formula, and graphing calculators. You can also use the equation of the parabola to find its x-intercepts, y-intercept, and other important features.

Q: What are some related topics to finding the equation of a parabola?

A: Some related topics to finding the equation of a parabola include:

• Quadratic Equations: Quadratic equations are a type of polynomial equation that can be represented in the form of $ax^2 + bx + c = 0$. The zeros of a quadratic equation are the points where the graph intersects the x-axis.
• Graphing Quadratic Equations: Quadratic equations can be graphed using various techniques, including factoring, the quadratic formula, and graphing calculators.
• Solving Systems of Equations: Systems of equations are a set of two or more equations that are solved simultaneously. Quadratic equations can be used to solve systems of equations.

Q: What are some practice problems related to finding the equation of a parabola?

A: Some practice problems related to finding the equation of a parabola include:

• Find the equation of a parabola that has zeros of $x = -1$ and $x = 4$ and a $y$-intercept of $(0, -20)$.
• Find the equation of a parabola that has zeros of $x = -3$ and $x = 2$ and a $y$-intercept of $(0, -15)$.
• Find the equation of a parabola that has zeros of $x = -5$ and $x = 1$ and a $y$-intercept of $(0, -25)$.

Conclusion

In this article, we have answered some frequently asked questions about finding the equation of a parabola. We have discussed the importance of zeros and y-intercept in a parabola, how to find the value of $a$ in the equation $y = ax^2 + bx + c$, and how to graph a parabola. We have also provided some related topics and practice problems to help you learn more about finding the equation of a parabola.