Find The Coordinates Of The Vertex For The Graph Of Y = X 2 + 8 X − 1 Y = X^2 + 8x - 1 Y = X 2 + 8 X − 1 .

Introduction

In mathematics, the vertex of a quadratic function is the highest or lowest point on the graph of the function. It is a crucial concept in algebra and calculus, and it has numerous applications in various fields such as physics, engineering, and economics. In this article, we will focus on finding the coordinates of the vertex for the graph of the quadratic function $y = x^2 + 8x - 1$.

What is a Quadratic Function?

A quadratic function is a polynomial function of degree two, which means that the highest power of the variable is two. The general form of a quadratic function is $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. The graph of a quadratic function is a parabola, which is a U-shaped curve that opens upwards or downwards.

The Vertex Form of a Quadratic Function

The vertex form of a quadratic function is $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. This form is useful for finding the coordinates of the vertex, as well as for graphing the parabola.

Finding the Coordinates of the Vertex

To find the coordinates of the vertex for the graph of $y = x^2 + 8x - 1$, we need to complete the square. This involves rewriting the quadratic function in the form $y = a(x - h)^2 + k$. We can do this by adding and subtracting a constant term inside the parentheses.

Step 1: Add and Subtract a Constant Term

We start by adding and subtracting $(8/2)^2 = 16$ inside the parentheses:

$$y = x^2 + 8x - 1 = (x^2 + 8x + 16) - 16 - 1$$

Step 2: Factor the Perfect Square

Now, we can factor the perfect square inside the parentheses:

$$y = (x + 4)^2 - 17$$

Step 3: Identify the Vertex

Comparing this expression with the vertex form of a quadratic function, we can see that the vertex is at the point $(-4, -17)$.

Conclusion

In this article, we have found the coordinates of the vertex for the graph of the quadratic function $y = x^2 + 8x - 1$. We have used the method of completing the square to rewrite the quadratic function in the vertex form, and then identified the vertex as the point $(-4, -17)$. This is a crucial concept in algebra and calculus, and it has numerous applications in various fields.

Example Problems

1. Find the coordinates of the vertex for the graph of $y = x^2 - 6x + 2$.
2. Find the coordinates of the vertex for the graph of $y = x^2 + 2x - 3$.
3. Find the coordinates of the vertex for the graph of $y = x^2 - 4x + 1$.

Solutions

1. To find the coordinates of the vertex for the graph of $y = x^2 - 6x + 2$, we can complete the square as follows:

$$y = x^2 - 6x + 2 = (x^2 - 6x + 9) - 9 + 2 = (x - 3)^2 - 7$$

The vertex is at the point $(3, -7)$.

2. To find the coordinates of the vertex for the graph of $y = x^2 + 2x - 3$, we can complete the square as follows:

$$y = x^2 + 2x - 3 = (x^2 + 2x + 1) - 1 - 3 = (x + 1)^2 - 4$$

The vertex is at the point $(-1, -4)$.

3. To find the coordinates of the vertex for the graph of $y = x^2 - 4x + 1$, we can complete the square as follows:

$$y = x^2 - 4x + 1 = (x^2 - 4x + 4) - 4 + 1 = (x - 2)^2 - 3$$

The vertex is at the point $(2, -3)$.

Tips and Tricks

• To find the coordinates of the vertex for a quadratic function, you can use the method of completing the square.
• The vertex form of a quadratic function is $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.
• The vertex of a parabola is the highest or lowest point on the graph of the function.
• The method of completing the square is a useful technique for finding the coordinates of the vertex for a quadratic function.
  Quadratic Function Vertex: Frequently Asked Questions =====================================================

Introduction

In our previous article, we discussed how to find the coordinates of the vertex for a quadratic function. In this article, we will answer some frequently asked questions about quadratic function vertices.

Q: What is the vertex of a quadratic function?

A: The vertex of a quadratic function is the highest or lowest point on the graph of the function. It is a crucial concept in algebra and calculus, and it has numerous applications in various fields.

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

A: To find the coordinates of the vertex for a quadratic function, you can use the method of completing the square. This involves rewriting the quadratic function in the form $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.

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

A: The vertex form of a quadratic function is $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. This form is useful for finding the coordinates of the vertex, as well as for graphing the parabola.

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

A: To complete the square, you need to add and subtract a constant term inside the parentheses. This involves rewriting the quadratic function in the form $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.

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

A: The vertex of a quadratic function is the highest or lowest point on the graph of the function. It is a crucial concept in algebra and calculus, and it has numerous applications in various fields.

Q: Can I use the vertex form of a quadratic function to graph the parabola?

A: Yes, you can use the vertex form of a quadratic function to graph the parabola. The vertex form is $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.

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

A: To find the x-coordinate of the vertex of a quadratic function, you can use the formula $h = -\frac{b}{2a}$, where $a$ and $b$ are the coefficients of the quadratic function.

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

A: To find the y-coordinate of the vertex of a quadratic function, you can substitute the x-coordinate into the quadratic function and solve for $y$.

Q: Can I use a calculator to find the vertex of a quadratic function?

A: Yes, you can use a calculator to find the vertex of a quadratic function. Most graphing calculators have a built-in function to find the vertex of a quadratic function.

Q: What are some common mistakes to avoid when finding the vertex of a quadratic function?

A: Some common mistakes to avoid when finding the vertex of a quadratic function include:

• Not completing the square correctly
• Not using the correct formula to find the x-coordinate of the vertex
• Not substituting the x-coordinate into the quadratic function to find the y-coordinate
• Not using a calculator to check the answer

Conclusion

In this article, we have answered some frequently asked questions about quadratic function vertices. We have discussed how to find the coordinates of the vertex for a quadratic function, as well as some common mistakes to avoid when finding the vertex. We hope that this article has been helpful in clarifying any confusion about quadratic function vertices.

Example Problems

1. Find the coordinates of the vertex for the graph of $y = x^2 - 6x + 2$.
2. Find the coordinates of the vertex for the graph of $y = x^2 + 2x - 3$.
3. Find the coordinates of the vertex for the graph of $y = x^2 - 4x + 1$.

Solutions

1. To find the coordinates of the vertex for the graph of $y = x^2 - 6x + 2$, we can complete the square as follows:

$$y = x^2 - 6x + 2 = (x^2 - 6x + 9) - 9 + 2 = (x - 3)^2 - 7$$

The vertex is at the point $(3, -7)$.

2. To find the coordinates of the vertex for the graph of $y = x^2 + 2x - 3$, we can complete the square as follows:

$$y = x^2 + 2x - 3 = (x^2 + 2x + 1) - 1 - 3 = (x + 1)^2 - 4$$

The vertex is at the point $(-1, -4)$.

3. To find the coordinates of the vertex for the graph of $y = x^2 - 4x + 1$, we can complete the square as follows:

$$y = x^2 - 4x + 1 = (x^2 - 4x + 4) - 4 + 1 = (x - 2)^2 - 3$$

The vertex is at the point $(2, -3)$.

Tips and Tricks

• To find the coordinates of the vertex for a quadratic function, you can use the method of completing the square.
• The vertex form of a quadratic function is $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.
• The vertex of a parabola is the highest or lowest point on the graph of the function.
• The method of completing the square is a useful technique for finding the coordinates of the vertex for a quadratic function.