An Image Of A Parabolic Lens Is Projected Onto A Graph. The \[$ Y \$\]-intercept Of The Graph Is \[$(0, 90)\$\], And The Zeros Are 5 And 9. Which Equation Models The Function?A. \[$ Y = 90(x-5)(x-9) \$\]B. \[$ Y =

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Introduction

In mathematics, parabolic equations are a fundamental concept in algebra and calculus. These equations are used to model various real-world phenomena, such as the trajectory of projectiles, the shape of mirrors, and the behavior of lenses. In this article, we will explore how to identify and model parabolic equations using a given graph.

The Graph: A Parabolic Lens

The graph in question is a parabolic lens, which is a type of conic section. The graph has a y-intercept of (0, 90), indicating that the graph passes through the point (0, 90) on the coordinate plane. The zeros of the graph are 5 and 9, which means that the graph intersects the x-axis at these points.

Identifying the Equation

To identify the equation that models the function, we need to use the information provided by the graph. The general form of a parabolic equation is:

y = a(x - h)(x - k)

where a is the coefficient, h is the x-coordinate of the vertex, and k is the y-coordinate of the vertex.

Using the Given Information

We are given that the y-intercept is (0, 90), which means that the vertex of the parabola is at (h, k) = (0, 90). We are also given that the zeros are 5 and 9, which means that the parabola intersects the x-axis at these points.

Finding the Equation

Using the given information, we can write the equation of the parabola as:

y = a(x - 5)(x - 9)

To find the value of a, we can use the fact that the y-intercept is (0, 90). Substituting x = 0 and y = 90 into the equation, we get:

90 = a(0 - 5)(0 - 9)

Simplifying the equation, we get:

90 = 45a

Dividing both sides by 45, we get:

a = 2

The Final Equation

Substituting the value of a into the equation, we get:

y = 2(x - 5)(x - 9)

This is the equation that models the function.

Conclusion

In this article, we have explored how to identify and model parabolic equations using a given graph. We have used the information provided by the graph to write the equation of the parabola and have found the value of the coefficient a. The final equation is:

y = 2(x - 5)(x - 9)

This equation models the function and can be used to describe the behavior of the parabolic lens.

Discussion Questions

  1. What is the general form of a parabolic equation?
  2. How do you find the value of the coefficient a in a parabolic equation?
  3. What is the significance of the y-intercept in a parabolic equation?
  4. How do you use the zeros of a parabolic equation to find the equation?
  5. What is the final equation that models the function?

Answer Key

  1. y = a(x - h)(x - k)
  2. You can find the value of a by using the fact that the y-intercept is (0, 90) and substituting x = 0 and y = 90 into the equation.
  3. The y-intercept is the point where the graph intersects the y-axis.
  4. You can use the zeros of a parabolic equation to find the equation by substituting the zeros into the equation and solving for a.
  5. y = 2(x - 5)(x - 9)

Introduction

In our previous article, we explored how to identify and model parabolic equations using a given graph. In this article, we will provide a comprehensive Q&A guide to help you better understand parabolic equations and how to work with them.

Q&A: Parabolic Equations

Q1: What is the general form of a parabolic equation?

A1: The general form of a parabolic equation is y = a(x - h)(x - k), where a is the coefficient, h is the x-coordinate of the vertex, and k is the y-coordinate of the vertex.

Q2: How do you find the value of the coefficient a in a parabolic equation?

A2: You can find the value of a by using the fact that the y-intercept is (0, 90) and substituting x = 0 and y = 90 into the equation. For example, if the y-intercept is (0, 90), you can substitute x = 0 and y = 90 into the equation y = a(x - h)(x - k) to get 90 = a(0 - h)(0 - k).

Q3: What is the significance of the y-intercept in a parabolic equation?

A3: The y-intercept is the point where the graph intersects the y-axis. It is an important point in a parabolic equation because it helps to determine the value of the coefficient a.

Q4: How do you use the zeros of a parabolic equation to find the equation?

A4: You can use the zeros of a parabolic equation to find the equation by substituting the zeros into the equation and solving for a. For example, if the zeros are 5 and 9, you can substitute x = 5 and x = 9 into the equation y = a(x - h)(x - k) to get 0 = a(5 - h)(5 - k) and 0 = a(9 - h)(9 - k).

Q5: What is the final equation that models the function?

A5: The final equation that models the function is y = 2(x - 5)(x - 9).

Q6: How do you determine the vertex of a parabolic equation?

A6: You can determine the vertex of a parabolic equation by using the fact that the vertex is the midpoint of the two zeros. For example, if the zeros are 5 and 9, the vertex is the midpoint of 5 and 9, which is (7, 0).

Q7: How do you use the vertex to find the equation of a parabolic equation?

A7: You can use the vertex to find the equation of a parabolic equation by substituting the vertex into the equation y = a(x - h)(x - k). For example, if the vertex is (7, 0), you can substitute x = 7 and y = 0 into the equation to get 0 = a(7 - h)(7 - k).

Q8: What is the significance of the coefficient a in a parabolic equation?

A8: The coefficient a determines the direction and width of the parabola. A positive value of a indicates that the parabola opens upward, while a negative value of a indicates that the parabola opens downward.

Q9: How do you graph a parabolic equation?

A9: You can graph a parabolic equation by using the zeros and the vertex to plot the points on the graph. For example, if the zeros are 5 and 9 and the vertex is (7, 0), you can plot the points (5, 0), (7, 0), and (9, 0) on the graph.

Q10: What are some common applications of parabolic equations?

A10: Parabolic equations have many common applications in physics, engineering, and mathematics. Some examples include the trajectory of projectiles, the shape of mirrors, and the behavior of lenses.

Conclusion

In this article, we have provided a comprehensive Q&A guide to help you better understand parabolic equations and how to work with them. We have covered topics such as the general form of a parabolic equation, finding the value of the coefficient a, and graphing a parabolic equation. We hope that this guide has been helpful in your studies of parabolic equations.

Discussion Questions

  1. What is the general form of a parabolic equation?
  2. How do you find the value of the coefficient a in a parabolic equation?
  3. What is the significance of the y-intercept in a parabolic equation?
  4. How do you use the zeros of a parabolic equation to find the equation?
  5. What is the final equation that models the function?

Answer Key

  1. y = a(x - h)(x - k)
  2. You can find the value of a by using the fact that the y-intercept is (0, 90) and substituting x = 0 and y = 90 into the equation.
  3. The y-intercept is the point where the graph intersects the y-axis.
  4. You can use the zeros of a parabolic equation to find the equation by substituting the zeros into the equation and solving for a.
  5. y = 2(x - 5)(x - 9)