Find The Vertex Of The Given Quadratic Function.$f(x) = (x-8)(x-4$\]A. $(6, -4$\] B. $(-6, 4$\] C. $(-4, 8$\] D. $(4, 8$\]

by ADMIN 128 views

Introduction

In algebra, 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 learn how to find the vertex of a given quadratic function.

What is the Vertex of a Quadratic Function?

The vertex of a quadratic function is the point on the graph where the function changes from decreasing to increasing or vice versa. It is the highest or lowest point on the graph, depending on the direction of the parabola. The vertex is denoted by the point (h, k), where h is the x-coordinate and k is the y-coordinate.

How to Find the Vertex of a Quadratic Function

To find the vertex of a quadratic function, we can use the formula:

h = -b / 2a

k = f(h)

where h is the x-coordinate of the vertex, k is the y-coordinate of the vertex, a is the coefficient of the x^2 term, and b is the coefficient of the x term.

Finding the Vertex of the Given Quadratic Function

The given quadratic function is f(x) = (x-8)(x-4). To find the vertex, we need to expand the function and rewrite it in the standard form.

f(x) = (x-8)(x-4) f(x) = x^2 - 4x - 8x + 32 f(x) = x^2 - 12x + 32

Now, we can identify the values of a and b.

a = 1 b = -12

Using the formula, we can find the x-coordinate of the vertex.

h = -b / 2a h = -(-12) / 2(1) h = 12 / 2 h = 6

Now, we need to find the y-coordinate of the vertex. We can do this by substituting the value of h into the function.

k = f(h) k = f(6) k = (6-8)(6-4) k = (-2)(2) k = -4

Therefore, the vertex of the given quadratic function is (6, -4).

Conclusion

In this article, we learned how to find the vertex of a quadratic function. We used the formula h = -b / 2a and k = f(h) to find the x-coordinate and y-coordinate of the vertex. We also applied this formula to the given quadratic function f(x) = (x-8)(x-4) and found the vertex to be (6, -4). This is a useful skill to have in algebra and is essential for understanding the behavior of quadratic functions.

Answer

The correct answer is A. (6, -4).

Other Options

The other options are:

B. (-6, 4) C. (-4, 8) D. (4, 8)

These options are incorrect because they do not match the vertex of the given quadratic function.

Tips and Tricks

  • To find the vertex of a quadratic function, you need to identify the values of a and b.
  • Use the formula h = -b / 2a to find the x-coordinate of the vertex.
  • Substitute the value of h into the function to find the y-coordinate of the vertex.
  • Make sure to simplify the expression and evaluate the function correctly.

Practice Problems

  • Find the vertex of the quadratic function f(x) = x^2 + 5x + 6.
  • Find the vertex of the quadratic function f(x) = x^2 - 3x - 4.
  • Find the vertex of the quadratic function f(x) = 2x^2 + x - 1.

Solutions

  • The vertex of the quadratic function f(x) = x^2 + 5x + 6 is (-5/2, 13/4).
  • The vertex of the quadratic function f(x) = x^2 - 3x - 4 is (3/2, -13/4).
  • The vertex of the quadratic function f(x) = 2x^2 + x - 1 is (-1/4, 17/16).

Conclusion

Q: What is the vertex of a quadratic function?

A: The vertex of a quadratic function is the maximum or minimum point on the graph of the function. It is the highest or lowest point on the graph, depending on the direction of the parabola.

Q: How do I find the vertex of a quadratic function?

A: To find the vertex of a quadratic function, you need to identify the values of a and b. Then, use the formula h = -b / 2a to find the x-coordinate of the vertex. Finally, substitute the value of h into the function to find the y-coordinate of the vertex.

Q: What is the formula for finding the vertex of a quadratic function?

A: The formula for finding the vertex of a quadratic function is:

h = -b / 2a k = f(h)

where h is the x-coordinate of the vertex, k is the y-coordinate of the vertex, a is the coefficient of the x^2 term, and b is the coefficient of the x term.

Q: How do I identify the values of a and b in a quadratic function?

A: To identify the values of a and b in a quadratic function, you need to rewrite the function in the standard form, ax^2 + bx + c. Then, the value of a is the coefficient of the x^2 term, and the value of b is the coefficient of the x term.

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

A: The vertex of a quadratic function is significant because it represents the maximum or minimum point on the graph of the function. It is also the point where the function changes from decreasing to increasing or vice versa.

Q: Can the vertex of a quadratic function be a maximum or a minimum?

A: Yes, the vertex of a quadratic function can be a maximum or a minimum. If the parabola opens upward, the vertex is a minimum point. If the parabola opens downward, the vertex is a maximum point.

Q: How do I determine whether the vertex of a quadratic function is a maximum or a minimum?

A: To determine whether the vertex of a quadratic function is a maximum or a minimum, you need to look at the direction of the parabola. If the parabola opens upward, the vertex is a minimum point. If the parabola opens downward, the vertex is a maximum point.

Q: Can the vertex of a quadratic function be a point of inflection?

A: No, the vertex of a quadratic function cannot be a point of inflection. A point of inflection is a point on the graph of a function where the function changes from concave to convex or vice versa. The vertex of a quadratic function is a point of maximum or minimum, not a point of inflection.

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 need to use the formula h = -b / 2a, just like for a quadratic function with a positive leading coefficient. However, the vertex will be a maximum point, not a minimum point.

Q: Can the vertex of a quadratic function be a point of discontinuity?

A: No, the vertex of a quadratic function cannot be a point of discontinuity. A point of discontinuity is a point on the graph of a function where the function is not defined. The vertex of a quadratic function is a point on the graph where the function is defined.

Q: How do I find the vertex of a quadratic function with a rational coefficient?

A: To find the vertex of a quadratic function with a rational coefficient, you need to use the formula h = -b / 2a, just like for a quadratic function with a rational coefficient. However, you may need to simplify the expression to find the x-coordinate of the vertex.

Q: Can the vertex of a quadratic function be a point of symmetry?

A: Yes, the vertex of a quadratic function can be a point of symmetry. The vertex is the point on the graph where the function is symmetric about the x-axis or the y-axis.

Q: How do I find the vertex of a quadratic function with a complex coefficient?

A: To find the vertex of a quadratic function with a complex coefficient, you need to use the formula h = -b / 2a, just like for a quadratic function with a real coefficient. However, you may need to use complex numbers to find the x-coordinate of the vertex.

Conclusion

In conclusion, finding the vertex of a quadratic function is an essential skill in algebra. By using the formula h = -b / 2a and k = f(h), we can find the x-coordinate and y-coordinate of the vertex. We also discussed various scenarios, such as quadratic functions with negative leading coefficients, rational coefficients, and complex coefficients.