The Quadratic Function F ( X ) = X 2 − 6 X + 5 F(x) = X^2 - 6x + 5 F ( X ) = X 2 − 6 X + 5 Can Be Rewritten In The Form F ( X ) = A ( X − H ) 2 + K F(x) = A(x-h)^2 + K F ( X ) = A ( X − H ) 2 + K . Find The Real Numbers A A A , H H H , And K K K . Also, Provide The Coordinates Of The Vertex. Answer Exactly.$a =

Introduction

In mathematics, quadratic functions are a fundamental concept that plays a crucial role in various fields, including algebra, geometry, and calculus. A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. In this article, we will focus on the quadratic function $f(x) = x^2 - 6x + 5$ and rewrite it in the form $f(x) = a(x-h)^2 + k$. Our objective is to find the real numbers $a$, $h$, and $k$ and provide the coordinates of the vertex.

Completing the Square Method

To rewrite the quadratic function in the desired form, we will use the completing the square method. This method involves manipulating the quadratic expression to create a perfect square trinomial. The general form of a quadratic function is $f(x) = ax^2 + bx + c$. To complete the square, we need to add and subtract a constant term that will make the expression a perfect square trinomial.

Step 1: Factor out the coefficient of $x^2$

The first step is to factor out the coefficient of $x^2$, which is 1 in this case. This will give us $f(x) = (x^2 - 6x) + 5$.

Step 2: Add and subtract the square of half the coefficient of $x$

Next, we need to add and subtract the square of half the coefficient of $x$. In this case, the coefficient of $x$ is -6, so we need to add and subtract $(-6/2)^2 = 9$. This will give us $f(x) = (x^2 - 6x + 9) - 9 + 5$.

Step 3: Simplify the expression

Now, we can simplify the expression by combining like terms. This will give us $f(x) = (x - 3)^2 - 4$.

Finding the Real Numbers $a$, $h$, and $k$

Comparing the rewritten expression with the desired form $f(x) = a(x-h)^2 + k$, we can see that $a = 1$, $h = 3$, and $k = -4$.

Vertex Coordinates

The vertex of a quadratic function is the point where the function changes from decreasing to increasing or vice versa. The vertex coordinates can be found using the formula $(h, k)$. In this case, the vertex coordinates are $(3, -4)$.

Conclusion

In this article, we have successfully rewritten the quadratic function $f(x) = x^2 - 6x + 5$ in the form $f(x) = a(x-h)^2 + k$. We have also found the real numbers $a$, $h$, and $k$ and provided the coordinates of the vertex. The completing the square method is a powerful tool for rewriting quadratic functions and finding their vertex coordinates.

Final Answer

The final answer is:

Frequently Asked Questions

In this article, we will address some of the most common questions related to quadratic functions, including the completing the square method, vertex coordinates, and more.

Q: What is the completing the square method?

A: The completing the square method is a technique used to rewrite a quadratic function in the form $f(x) = a(x-h)^2 + k$. This method involves manipulating the quadratic expression to create a perfect square trinomial.

Q: How do I use the completing the square method?

A: To use the completing the square method, follow these steps:

1. Factor out the coefficient of $x^2$.
2. Add and subtract the square of half the coefficient of $x$.
3. Simplify the expression by combining like terms.

Q: What is the vertex of a quadratic function?

A: The vertex of a quadratic function is the point where the function changes from decreasing to increasing or vice versa. The vertex coordinates can be found using the formula $(h, k)$.

Q: How do I find the vertex coordinates?

A: To find the vertex coordinates, use the formula $(h, k)$. The value of $h$ is the x-coordinate of the vertex, and the value of $k$ is the y-coordinate of the vertex.

Q: What is the significance of the vertex coordinates?

A: The vertex coordinates are significant because they provide information about the behavior of the quadratic function. The vertex coordinates can be used to determine the maximum or minimum value of the function, as well as the direction of the function's increase or decrease.

Q: Can I use the completing the square method to find the vertex coordinates?

A: Yes, you can use the completing the square method to find the vertex coordinates. By rewriting the quadratic function in the form $f(x) = a(x-h)^2 + k$, you can easily identify the vertex coordinates as $(h, k)$.

Q: What are some common applications of quadratic functions?

A: Quadratic functions have many common applications in various fields, including:

• Physics: Quadratic functions are used to model the motion of objects under the influence of gravity or other forces.
• Engineering: Quadratic functions are used to design and optimize systems, such as bridges and buildings.
• Economics: Quadratic functions are used to model the behavior of economic systems, including supply and demand curves.

Q: How do I graph a quadratic function?

A: To graph a quadratic function, follow these steps:

1. Identify the vertex coordinates.
2. Determine the direction of the function's increase or decrease.
3. Plot the vertex point on the graph.
4. Plot additional points on the graph to create a smooth curve.

Conclusion

In this article, we have addressed some of the most common questions related to quadratic functions, including the completing the square method, vertex coordinates, and more. We hope that this article has provided you with a better understanding of quadratic functions and their applications.

Final Answer

The final answer is:

• The completing the square method is a technique used to rewrite a quadratic function in the form $f(x) = a(x-h)^2 + k$.
• The vertex coordinates can be found using the formula $(h, k)$.
• Quadratic functions have many common applications in various fields, including physics, engineering, and economics.
• To graph a quadratic function, identify the vertex coordinates, determine the direction of the function's increase or decrease, plot the vertex point, and plot additional points to create a smooth curve.