Given Information and the Equation of the Parabola

Oct 13, 2025 by miodeya.com 147 views

Understanding the Basics of a Parabola

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 the equation of the parabola can be written in factored form as $y = a(x - r)(x - s)$ , where $r$ and $s$ are the zeros of the parabola.

Given Information and the Equation of the Parabola

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 equation of the parabola can be written in factored form as $y = a(x + 4)(x - 2)$ .

Substituting the Given Point into the Equation

To determine the value of $a$ , we can substitute the given point (6, 10) into the equation of the parabola. This will give us an equation that we can solve for $a$ .

Option A: $10 = a(6-4)(6+2)$

Option A is $10 = a(6-4)(6+2)$ . Let's evaluate the expression inside the parentheses first. We have $(6-4) = 2$ and $(6+2) = 8$ . Therefore, the equation becomes $10 = a(2)(8)$ .

Simplifying the Equation

Simplifying the equation, we get $10 = 16a$ . To solve for $a$ , we can divide both sides of the equation by 16. This gives us $a = \frac{10}{16}$ .

Option B: $10 = a(6+4)(6-2)$

Option B is $10 = a(6+4)(6-2)$ . Let's evaluate the expression inside the parentheses first. We have $(6+4) = 10$ and $(6-2) = 4$ . Therefore, the equation becomes $10 = a(10)(4)$ .

Simplifying the Equation

Simplifying the equation, we get $10 = 40a$ . To solve for $a$ , we can divide both sides of the equation by 40. This gives us $a = \frac{10}{40}$ .

Option C: $6 = a(6+4)(6-2)$

Option C is $6 = a(6+4)(6-2)$ . Let's evaluate the expression inside the parentheses first. We have $(6+4) = 10$ and $(6-2) = 4$ . Therefore, the equation becomes $6 = a(10)(4)$ .

Simplifying the Equation

Simplifying the equation, we get $6 = 40a$ . To solve for $a$ , we can divide both sides of the equation by 40. This gives us $a = \frac{6}{40}$ .

Conclusion

In conclusion, we have three options to determine the value of $a$ in the equation of the parabola. Option A gives us $a = \frac{10}{16}$ , Option B gives us $a = \frac{10}{40}$ , and Option C gives us $a = \frac{6}{40}$ . However, we need to check if these values satisfy the given point (6, 10) on the graph.

Checking the Values of $a$

Let's substitute the values of $a$ into the equation of the parabola and check if the point (6, 10) satisfies the equation.

Option A: $10 = a(6-4)(6+2)$

Substituting $a = \frac{10}{16}$ into the equation, we get $10 = \frac{10}{16}(2)(8)$ . Simplifying the equation, we get $10 = \frac{160}{16}$ , which is true.

Option B: $10 = a(6+4)(6-2)$

Substituting $a = \frac{10}{40}$ into the equation, we get $10 = \frac{10}{40}(10)(4)$ . Simplifying the equation, we get $10 = \frac{400}{40}$ , which is true.

Option C: $6 = a(6+4)(6-2)$

Substituting $a = \frac{6}{40}$ into the equation, we get $6 = \frac{6}{40}(10)(4)$ . Simplifying the equation, we get $6 = \frac{240}{40}$ , which is not true.

Conclusion

In conclusion, we have checked the values of $a$ in the equation of the parabola and found that Option A and Option B satisfy the given point (6, 10) on the graph. However, Option C does not satisfy the point.

Final Answer

Therefore, the correct equation that can be solved to determine the value of $a$ in the equation of the parabola is Option A: $10 = a(6-4)(6+2)$ or Option B: $10 = a(6+4)(6-2)$ .

Q: What is a parabola?

A: A parabola is a quadratic equation that can be represented in the form of $y = ax^2 + bx + c$ . It is a U-shaped curve that opens upwards or downwards.

Q: What are the zeros of a parabola?

A: The zeros of a parabola are the points where the graph intersects the x-axis. They are also known as the roots or solutions of the equation.

Q: How do I find the equation of a parabola?

A: To find the equation of a parabola, you need to know the zeros of the parabola and a point on the graph. You can use the factored form of the equation $y = a(x - r)(x - s)$ , where $r$ and $s$ are the zeros of the parabola.

Q: How do I determine the value of $a$ in the equation of a parabola?

A: To determine the value of $a$ in the equation of a parabola, you can substitute a point on the graph into the equation and solve for $a$ . You can also use the fact that the parabola passes through the point $(x, y)$ .

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

A: The value of $a$ in the equation of a parabola determines the shape and orientation of the parabola. A positive value of $a$ indicates that the parabola opens upwards, while a negative value of $a$ indicates that the parabola opens downwards.

Q: Can a parabola have more than two zeros?

A: No, a parabola can have at most two zeros. This is because the equation of a parabola is a quadratic equation, and quadratic equations have at most two solutions.

Q: Can a parabola have no zeros?

A: Yes, a parabola can have no zeros. This occurs when the equation of the parabola has no real solutions, such as when the discriminant is negative.

Q: How do I graph a parabola?

A: To graph a parabola, you can use the equation of the parabola and plot points on the graph. You can also use the fact that the parabola passes through the point $(x, y)$ .

Q: What are some real-world applications of parabolas?

A: Parabolas have many real-world applications, including:

Projectile motion: The trajectory of a projectile, such as a thrown ball or a rocket, follows a parabolic path.
Optics: The shape of a mirror or a lens can be described by a parabola.
Architecture: The shape of a dome or a vault can be described by a parabola.
Engineering: The shape of a bridge or a beam can be described by a parabola.

Q: Can I use a calculator to graph a parabola?

A: Yes, you can use a calculator to graph a parabola. Most graphing calculators have a built-in function to graph parabolas.

Q: Can I use a computer to graph a parabola?

A: Yes, you can use a computer to graph a parabola. Most computer software, such as graphing software or programming languages, have a built-in function to graph parabolas.

Q: How do I find the vertex of a parabola?

A: To find the vertex of a parabola, you can use the formula $x = -\frac{b}{2a}$ , where $a$ and $b$ are the coefficients of the quadratic equation.

Q: How do I find the axis of symmetry of a parabola?

A: To find the axis of symmetry of a parabola, you can use the formula $x = -\frac{b}{2a}$ , where $a$ and $b$ are the coefficients of the quadratic equation.

Q: Can I use a parabola to model real-world data?

A: Yes, you can use a parabola to model real-world data. Parabolas are often used to model data that has a quadratic relationship between the variables.

Q: How do I determine the best fit parabola for a set of data?

A: To determine the best fit parabola for a set of data, you can use a method such as least squares regression or a software package that can perform curve fitting.

Q: Can I use a parabola to model a non-quadratic relationship?

A: No, a parabola can only be used to model a quadratic relationship. If the relationship is non-quadratic, you may need to use a different type of curve, such as a polynomial or a trigonometric function.

Q: How do I find the equation of a parabola given three points?

A: To find the equation of a parabola given three points, you can use a method such as Lagrange interpolation or a software package that can perform curve fitting.

Q: Can I use a parabola to model a periodic relationship?

A: No, a parabola is not suitable for modeling periodic relationships. If the relationship is periodic, you may need to use a different type of curve, such as a sine or cosine function.

Q: How do I find the equation of a parabola given a point and a tangent line?

A: To find the equation of a parabola given a point and a tangent line, you can use a method such as implicit differentiation or a software package that can perform curve fitting.

Q: Can I use a parabola to model a relationship with a discontinuity?

A: No, a parabola is not suitable for modeling relationships with discontinuities. If the relationship has a discontinuity, you may need to use a different type of curve, such as a piecewise function.

Q: How do I find the equation of a parabola given a point and a normal line?

A: To find the equation of a parabola given a point and a normal line, you can use a method such as implicit differentiation or a software package that can perform curve fitting.

Q: Can I use a parabola to model a relationship with a singularity?

A: No, a parabola is not suitable for modeling relationships with singularities. If the relationship has a singularity, you may need to use a different type of curve, such as a rational function.

Q: How do I find the equation of a parabola given a point and a secant line?

A: To find the equation of a parabola given a point and a secant line, you can use a method such as implicit differentiation or a software package that can perform curve fitting.

Q: Can I use a parabola to model a relationship with a periodicity?

A: No, a parabola is not suitable for modeling relationships with periodicity. If the relationship has periodicity, you may need to use a different type of curve, such as a sine or cosine function.

Q: How do I find the equation of a parabola given a point and a radius of curvature?

A: To find the equation of a parabola given a point and a radius of curvature, you can use a method such as implicit differentiation or a software package that can perform curve fitting.

Q: Can I use a parabola to model a relationship with a curvature?

A: No, a parabola is not suitable for modeling relationships with curvature. If the relationship has curvature, you may need to use a different type of curve, such as a cubic or quartic function.

Q: How do I find the equation of a parabola given a point and a curvature?

A: To find the equation of a parabola given a point and a curvature, you can use a method such as implicit differentiation or a software package that can perform curve fitting.

Q: Can I use a parabola to model a relationship with a torsion?

A: No, a parabola is not suitable for modeling relationships with torsion. If the relationship has torsion, you may need to use a different type of curve, such as a helix or a spiral.

Q: How do I find the equation of a parabola given a point and a torsion?

A: To find the equation of a parabola given a point and a torsion, you can use a method such as implicit differentiation or a software package that can perform curve fitting.

Q: Can I use a parabola to model a relationship with a twist?

A: No, a parabola is not suitable for modeling relationships with twist. If the relationship has twist, you may need to use a different type of curve, such as a helix or a spiral.

Q: How do I find the equation of a parabola given a point and a twist?

A: To find the equation of a parabola given a point and a twist, you can use a method such as implicit differentiation or a software package that can perform curve fitting.

Q: Can I use a parabola to model a relationship with a symmetry?

A: Yes, a parabola can be used to model relationships with symmetry. A parabola has an axis