What Is The Equation Of The Quadratic Function With A Vertex At { (2, -25)$}$ And An { X$}$-intercept At { (7, 0)$}$?A. { F(x) = (x-2)(x-7)$}$B. { F(x) = (x+2)(x+7)$} C . \[ C. \[ C . \[ F(x) =
What is the Equation of the Quadratic Function with a Vertex at (2, -25) and an x-Intercept at (7, 0)?
Understanding Quadratic Functions
Quadratic functions are a type of polynomial function that can be written in the form f(x) = ax^2 + bx + c, where a, b, and c are constants. These functions have a parabolic shape and can be represented graphically as a U-shaped curve. Quadratic functions have a wide range of applications in various fields, including physics, engineering, and economics.
Vertex Form of a Quadratic Function
The vertex form of a quadratic function is given by f(x) = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. In this form, the vertex is the point on the graph where the function changes direction. The vertex form is particularly useful when working with quadratic functions that have a known vertex.
Given Information
We are given that the vertex of the quadratic function is at (2, -25) and the x-intercept is at (7, 0). This means that the function passes through the points (2, -25) and (7, 0).
Finding the Equation of the Quadratic Function
To find the equation of the quadratic function, we can use the vertex form and the given information. Since the vertex is at (2, -25), we can write the equation as f(x) = a(x - 2)^2 - 25.
Using the x-Intercept to Find the Value of a
We are also given that the x-intercept is at (7, 0). This means that when x = 7, f(x) = 0. We can substitute these values into the equation to find the value of a.
f(7) = a(7 - 2)^2 - 25 0 = a(5)^2 - 25 0 = 25a - 25 25a = 25 a = 1
Writing the Equation of the Quadratic Function
Now that we have found the value of a, we can write the equation of the quadratic function.
f(x) = (x - 2)^2 - 25 f(x) = (x^2 - 4x + 4) - 25 f(x) = x^2 - 4x - 21
Comparing with the Given Options
We can compare our equation with the given options to see which one matches.
A. f(x) = (x - 2)(x - 7) B. f(x) = (x + 2)(x + 7) C. f(x) = (x - 2)(x - 7)
Our equation matches option A.
Conclusion
In conclusion, the equation of the quadratic function with a vertex at (2, -25) and an x-intercept at (7, 0) is f(x) = (x - 2)(x - 7). This equation represents a parabola that passes through the given points and has a vertex at (2, -25).
Key Takeaways
- The vertex form of a quadratic function is given by f(x) = a(x - h)^2 + k.
- The vertex form is particularly useful when working with quadratic functions that have a known vertex.
- To find the equation of a quadratic function, we can use the vertex form and the given information.
- We can use the x-intercept to find the value of a in the equation.
Final Answer
The final answer is A. f(x) = (x - 2)(x - 7).
Quadratic Function Equation Q&A
Understanding Quadratic Functions
Quadratic functions are a type of polynomial function that can be written in the form f(x) = ax^2 + bx + c, where a, b, and c are constants. These functions have a parabolic shape and can be represented graphically as a U-shaped curve. Quadratic functions have a wide range of applications in various fields, including physics, engineering, and economics.
Frequently Asked Questions
Q: What is the vertex form of a quadratic function?
A: The vertex form of a quadratic function is given by f(x) = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.
Q: How do I find the equation of a quadratic function with a known vertex?
A: To find the equation of a quadratic function with a known vertex, you can use the vertex form and the given information. You can write the equation as f(x) = a(x - h)^2 + k, where (h, k) is the vertex.
Q: How do I use the x-intercept to find the value of a in the equation?
A: To use the x-intercept to find the value of a, you can substitute the x-intercept into the equation and solve for a. For example, if the x-intercept is at (7, 0), you can substitute x = 7 and f(x) = 0 into the equation to find the value of a.
Q: What is the equation of a quadratic function with a vertex at (2, -25) and an x-intercept at (7, 0)?
A: The equation of a quadratic function with a vertex at (2, -25) and an x-intercept at (7, 0) is f(x) = (x - 2)(x - 7).
Q: How do I compare the equation of a quadratic function with the given options?
A: To compare the equation of a quadratic function with the given options, you can substitute the x-intercept into the equation and solve for a. You can then compare the resulting equation with the given options to see which one matches.
Q: What are some key takeaways when working with quadratic functions?
A: Some key takeaways when working with quadratic functions include:
- The vertex form of a quadratic function is given by f(x) = a(x - h)^2 + k.
- The vertex form is particularly useful when working with quadratic functions that have a known vertex.
- To find the equation of a quadratic function, you can use the vertex form and the given information.
- You can use the x-intercept to find the value of a in the equation.
Q: What is the final answer to the problem?
A: The final answer to the problem is A. f(x) = (x - 2)(x - 7).
Conclusion
In conclusion, quadratic functions are a type of polynomial function that can be written in the form f(x) = ax^2 + bx + c. The vertex form of a quadratic function is given by f(x) = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. To find the equation of a quadratic function with a known vertex, you can use the vertex form and the given information. You can use the x-intercept to find the value of a in the equation. The final answer to the problem is A. f(x) = (x - 2)(x - 7).
Key Takeaways
- The vertex form of a quadratic function is given by f(x) = a(x - h)^2 + k.
- The vertex form is particularly useful when working with quadratic functions that have a known vertex.
- To find the equation of a quadratic function, you can use the vertex form and the given information.
- You can use the x-intercept to find the value of a in the equation.
Final Answer
The final answer is A. f(x) = (x - 2)(x - 7).