Graph A Parabola Whose Vertex Is At ( 3 , 5 (3, 5 ( 3 , 5 ] With A Y Y Y -intercept At Y = 1 Y = 1 Y = 1 .

Mar 12, 2025 by ADMIN 107 views

**Graphing a Parabola with a Given Vertex and Y-Intercept**

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Introduction

In this article, we will explore the process of graphing a parabola with a given vertex and y-intercept. A parabola is a quadratic function that can be represented in the form of $y = ax^2 + bx + c$ . The vertex of a parabola is the point where the parabola changes direction, and it is represented by the coordinates $(h, k)$ . The y-intercept of a parabola is the point where the parabola intersects the y-axis, and it is represented by the coordinates $(0, c)$ .

Understanding the Problem

The problem requires us to graph a parabola with a vertex at $(3, 5)$ and a y-intercept at $y = 1$ . This means that the parabola will have a minimum or maximum point at $(3, 5)$ , and it will intersect the y-axis at the point $(0, 1)$ .

Finding the Equation of the Parabola

To find the equation of the parabola, we can use the vertex form of a quadratic function, which is represented by the equation $y = a(x - h)^2 + k$ . In this case, the vertex is at $(3, 5)$ , so we can substitute these values into the equation to get $y = a(x - 3)^2 + 5$ .

Using the Y-Intercept to Find the Value of a

The y-intercept of the parabola is at $(0, 1)$ , which means that when $x = 0$ , $y = 1$ . We can substitute these values into the equation to get $1 = a(0 - 3)^2 + 5$ . Simplifying this equation, we get $1 = 9a + 5$ . Subtracting 5 from both sides, we get $-4 = 9a$ . Dividing both sides by 9, we get $a = -\frac{4}{9}$ .

Writing the Equation of the Parabola

Now that we have found the value of $a$ , we can write the equation of the parabola. Substituting $a = -\frac{4}{9}$ into the equation $y = a(x - 3)^2 + 5$ , we get $y = -\frac{4}{9}(x - 3)^2 + 5$ .

Graphing the Parabola

To graph the parabola, we can use the equation $y = -\frac{4}{9}(x - 3)^2 + 5$ . We can start by plotting the vertex at $(3, 5)$ . Then, we can use the equation to find the y-coordinates of the parabola at various x-coordinates. For example, when $x = 0$ , $y = -\frac{4}{9}(0 - 3)^2 + 5 = 1$ . When $x = 6$ , $y = -\frac{4}{9}(6 - 3)^2 + 5 = 9$ . We can continue this process to find the y-coordinates of the parabola at various x-coordinates.

Conclusion

In this article, we have explored the process of graphing a parabola with a given vertex and y-intercept. We have used the vertex form of a quadratic function to find the equation of the parabola, and we have used the y-intercept to find the value of $a$ . We have then written the equation of the parabola and graphed it using the equation.

Example Problems

Problem 1

Graph a parabola with a vertex at $(2, 4)$ and a y-intercept at $y = 2$ .

Solution

To solve this problem, we can use the vertex form of a quadratic function, which is represented by the equation $y = a(x - h)^2 + k$ . In this case, the vertex is at $(2, 4)$ , so we can substitute these values into the equation to get $y = a(x - 2)^2 + 4$ . We can then use the y-intercept to find the value of $a$ . The y-intercept is at $(0, 2)$ , so we can substitute these values into the equation to get $2 = a(0 - 2)^2 + 4$ . Simplifying this equation, we get $2 = 4a + 4$ . Subtracting 4 from both sides, we get $-2 = 4a$ . Dividing both sides by 4, we get $a = -\frac{1}{2}$ . We can then write the equation of the parabola as $y = -\frac{1}{2}(x - 2)^2 + 4$ .

Problem 2

Graph a parabola with a vertex at $(1, 3)$ and a y-intercept at $y = 0$ .

Solution

To solve this problem, we can use the vertex form of a quadratic function, which is represented by the equation $y = a(x - h)^2 + k$ . In this case, the vertex is at $(1, 3)$ , so we can substitute these values into the equation to get $y = a(x - 1)^2 + 3$ . We can then use the y-intercept to find the value of $a$ . The y-intercept is at $(0, 0)$ , so we can substitute these values into the equation to get $0 = a(0 - 1)^2 + 3$ . Simplifying this equation, we get $0 = a + 3$ . Subtracting 3 from both sides, we get $-3 = a$ . We can then write the equation of the parabola as $y = -3(x - 1)^2 + 3$ .

Final Thoughts

Graphing a parabola with a given vertex and y-intercept requires us to use the vertex form of a quadratic function and the y-intercept to find the value of $a$ . We can then write the equation of the parabola and graph it using the equation. This process can be applied to a wide range of problems, and it is an important tool for graphing quadratic functions.

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Q: What is the vertex form of a quadratic function?

A: The vertex form of a quadratic function is represented by the equation $y = a(x - h)^2 + k$ , where $(h, k)$ is the vertex of the parabola.

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

A: To find the equation of a parabola with a given vertex and y-intercept, you can use the vertex form of a quadratic function. Substitute the vertex coordinates into the equation to get $y = a(x - h)^2 + k$ . Then, use the y-intercept to find the value of $a$ .

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

A: To find the value of $a$ in the equation of a parabola, you can use the y-intercept. Substitute the y-intercept coordinates into the equation to get $y = a(x - h)^2 + k$ . Then, solve for $a$ .

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

A: The y-intercept is the point where the parabola intersects the y-axis. It is represented by the coordinates $(0, c)$ , where $c$ is the y-coordinate of the y-intercept.

Q: How do I graph a parabola with a given vertex and y-intercept?

A: To graph a parabola with a given vertex and y-intercept, you can use the equation of the parabola. Plot the vertex and then use the equation to find the y-coordinates of the parabola at various x-coordinates.

Q: Can I graph a parabola with a given vertex and y-intercept using a graphing calculator?

A: Yes, you can graph a parabola with a given vertex and y-intercept using a graphing calculator. Enter the equation of the parabola and the vertex coordinates into the calculator, and it will graph the parabola for you.

Q: What are some common mistakes to avoid when graphing a parabola with a given vertex and y-intercept?

A: Some common mistakes to avoid when graphing a parabola with a given vertex and y-intercept include:

Not using the correct equation of the parabola
Not finding the correct value of $a$
Not plotting the vertex correctly
Not using the equation to find the y-coordinates of the parabola at various x-coordinates

Q: How do I check my work when graphing a parabola with a given vertex and y-intercept?

A: To check your work when graphing a parabola with a given vertex and y-intercept, you can:

Verify that the vertex is plotted correctly
Verify that the y-intercept is plotted correctly
Verify that the equation of the parabola is correct
Verify that the graph of the parabola is correct

Q: What are some real-world applications of graphing a parabola with a given vertex and y-intercept?

A: Some real-world applications of graphing a parabola with a given vertex and y-intercept include:

Modeling the trajectory of a projectile
Modeling the motion of an object under the influence of gravity
Modeling the growth of a population
Modeling the spread of a disease

Q: How do I extend my knowledge of graphing a parabola with a given vertex and y-intercept to more complex problems?

A: To extend your knowledge of graphing a parabola with a given vertex and y-intercept to more complex problems, you can:

Learn about more advanced topics in algebra, such as systems of equations and matrices
Learn about more advanced topics in calculus, such as optimization and differential equations
Practice solving more complex problems in graphing a parabola with a given vertex and y-intercept
Seek out additional resources, such as textbooks and online tutorials, to help you learn more about graphing a parabola with a given vertex and y-intercept.