Select The Correct Answer.What Is The Vertex Of The Quadratic Function $f(x)=(x-8)(x-4$\]?A. $(-4,8$\] B. $(8,-4$\] C. $(4,8$\] D. $(-6,4$\]
In mathematics, a quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The general form of a quadratic function is f(x) = ax^2 + bx + c, where a, b, and c are constants. The vertex of a quadratic function is the maximum or minimum point on the graph of the function. In this article, we will discuss how to find the vertex of a quadratic function and apply this concept to a specific problem.
What is the Vertex of a Quadratic Function?
The vertex of a quadratic function is the point at which the function changes from decreasing to increasing or vice versa. It is the minimum or maximum point on the graph of the function. The vertex can be found using the formula x = -b/2a, where a and b are the coefficients of the quadratic function. Once we have the x-coordinate of the vertex, we can find the y-coordinate by plugging the x-coordinate into the function.
Finding the Vertex of a Quadratic Function
To find the vertex of a quadratic function, we need to follow these steps:
- Write the quadratic function in the form f(x) = ax^2 + bx + c.
- Identify the coefficients a and b.
- Use the formula x = -b/2a to find the x-coordinate of the vertex.
- Plug the x-coordinate into the function to find the y-coordinate.
Applying the Concept to a Specific Problem
Now, let's apply the concept of finding the vertex of a quadratic function to the problem given in the discussion category.
Problem: What is the vertex of the quadratic function f(x) = (x - 8)(x - 4)?
Solution:
To find the vertex of the quadratic function f(x) = (x - 8)(x - 4), we need to expand the function and write it in the form f(x) = ax^2 + bx + c.
Step 1: Expand the Function
f(x) = (x - 8)(x - 4) = x^2 - 4x - 8x + 32 = x^2 - 12x + 32
Step 2: Identify the Coefficients
In the expanded function f(x) = x^2 - 12x + 32, the coefficients are a = 1 and b = -12.
Step 3: Find the X-Coordinate of the Vertex
Using the formula x = -b/2a, we can find the x-coordinate of the vertex.
x = -(-12)/2(1) = 12/2 = 6
Step 4: Find the Y-Coordinate of the Vertex
Now that we have the x-coordinate of the vertex, we can plug it into the function to find the y-coordinate.
f(6) = (6)^2 - 12(6) + 32 = 36 - 72 + 32 = -4
Conclusion:
Therefore, the vertex of the quadratic function f(x) = (x - 8)(x - 4) is (6, -4).
Answer:
In the previous article, we discussed how to find the vertex of a quadratic function. In this article, we will provide a comprehensive guide to quadratic function vertices, including a Q&A section to help you better understand the concept.
What is a Quadratic Function?
A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The general form of a quadratic function is f(x) = ax^2 + bx + c, where a, b, and c are constants.
What is the Vertex of a Quadratic Function?
The vertex of a quadratic function is the maximum or minimum point on the graph of the function. It is the point at which the function changes from decreasing to increasing or vice versa.
How to Find the Vertex of a Quadratic Function?
To find the vertex of a quadratic function, you need to follow these steps:
- Write the quadratic function in the form f(x) = ax^2 + bx + c.
- Identify the coefficients a and b.
- Use the formula x = -b/2a to find the x-coordinate of the vertex.
- Plug the x-coordinate into the function to find the y-coordinate.
Q&A: Quadratic Function Vertices
Q: What is the vertex of the quadratic function f(x) = x^2 + 4x + 4?
A: To find the vertex of the quadratic function f(x) = x^2 + 4x + 4, we need to follow the steps outlined above.
Step 1: Identify the Coefficients
In the quadratic function f(x) = x^2 + 4x + 4, the coefficients are a = 1 and b = 4.
Step 2: Find the X-Coordinate of the Vertex
Using the formula x = -b/2a, we can find the x-coordinate of the vertex.
x = -4/2(1) = -4/2 = -2
Step 3: Find the Y-Coordinate of the Vertex
Now that we have the x-coordinate of the vertex, we can plug it into the function to find the y-coordinate.
f(-2) = (-2)^2 + 4(-2) + 4 = 4 - 8 + 4 = 0
Conclusion:
Therefore, the vertex of the quadratic function f(x) = x^2 + 4x + 4 is (-2, 0).
Q: What is the vertex of the quadratic function f(x) = -x^2 + 6x - 9?
A: To find the vertex of the quadratic function f(x) = -x^2 + 6x - 9, we need to follow the steps outlined above.
Step 1: Identify the Coefficients
In the quadratic function f(x) = -x^2 + 6x - 9, the coefficients are a = -1 and b = 6.
Step 2: Find the X-Coordinate of the Vertex
Using the formula x = -b/2a, we can find the x-coordinate of the vertex.
x = -6/2(-1) = -6/-2 = 3
Step 3: Find the Y-Coordinate of the Vertex
Now that we have the x-coordinate of the vertex, we can plug it into the function to find the y-coordinate.
f(3) = -(3)^2 + 6(3) - 9 = -9 + 18 - 9 = 0
Conclusion:
Therefore, the vertex of the quadratic function f(x) = -x^2 + 6x - 9 is (3, 0).
Q: How do I find the vertex of a quadratic function with a negative leading coefficient?
A: To find the vertex of a quadratic function with a negative leading coefficient, you can use the formula x = -b/2a, just like you would for a quadratic function with a positive leading coefficient.
Example:
Suppose we want to find the vertex of the quadratic function f(x) = -x^2 + 4x - 3.
Step 1: Identify the Coefficients
In the quadratic function f(x) = -x^2 + 4x - 3, the coefficients are a = -1 and b = 4.
Step 2: Find the X-Coordinate of the Vertex
Using the formula x = -b/2a, we can find the x-coordinate of the vertex.
x = -4/2(-1) = -4/-2 = 2
Step 3: Find the Y-Coordinate of the Vertex
Now that we have the x-coordinate of the vertex, we can plug it into the function to find the y-coordinate.
f(2) = -(2)^2 + 4(2) - 3 = -4 + 8 - 3 = 1
Conclusion:
Therefore, the vertex of the quadratic function f(x) = -x^2 + 4x - 3 is (2, 1).
Q: Can I use the vertex formula to find the vertex of a quadratic function in the form f(x) = a(x - h)^2 + k?
A: Yes, you can use the vertex formula to find the vertex of a quadratic function in the form f(x) = a(x - h)^2 + k.
Example:
Suppose we want to find the vertex of the quadratic function f(x) = 2(x - 1)^2 + 3.
Step 1: Identify the Coefficients
In the quadratic function f(x) = 2(x - 1)^2 + 3, the coefficients are a = 2 and h = 1.
Step 2: Find the X-Coordinate of the Vertex
Using the formula x = h, we can find the x-coordinate of the vertex.
x = 1
Step 3: Find the Y-Coordinate of the Vertex
Now that we have the x-coordinate of the vertex, we can plug it into the function to find the y-coordinate.
f(1) = 2(1 - 1)^2 + 3 = 2(0)^2 + 3 = 3
Conclusion:
Therefore, the vertex of the quadratic function f(x) = 2(x - 1)^2 + 3 is (1, 3).
Conclusion:
In this article, we provided a comprehensive guide to quadratic function vertices, including a Q&A section to help you better understand the concept. We discussed how to find the vertex of a quadratic function using the formula x = -b/2a and how to use the vertex formula to find the vertex of a quadratic function in the form f(x) = a(x - h)^2 + k. We also provided examples to help illustrate the concept.