Step 2: Find The Vertex Using The Formula X = − B 2 A X = \frac{-b}{2a} X = 2 A − B .Calculate The X-coordinate Of The Vertex:${ X = \frac{4}{2(1)} }$ { X = 2 \} Find The Y-coordinate Of The Vertex:${ F(2) = 2^2 - 4(2) - 5 }$[

Mar 9, 2025 by ADMIN 233 views

**Step 2: Find the Vertex of a Quadratic Function**

Introduction

In the previous step, we learned how to write the equation of a quadratic function in the form $f(x) = ax^2 + bx + c$ . Now, we will learn how to find the vertex of a quadratic function using the formula $x = \frac{-b}{2a}$ . The vertex of a quadratic function is the maximum or minimum point on the graph of the 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 from increasing to decreasing. It is the highest or lowest point on the graph of the function. The vertex is denoted by the point $(h, k)$ , where $h$ is the x-coordinate of the vertex and $k$ is the y-coordinate of the vertex.

Finding the X-Coordinate of the Vertex

To find the x-coordinate of the vertex, we use the formula $x = \frac{-b}{2a}$ . This formula is derived from the equation of the quadratic function in the form $f(x) = ax^2 + bx + c$ . By substituting the values of $a$ and $b$ into the formula, we can find the x-coordinate of the vertex.

Example: Finding the X-Coordinate of the Vertex

Let's consider the quadratic function $f(x) = x^2 + 4x + 5$ . To find the x-coordinate of the vertex, we need to substitute the values of $a$ and $b$ into the formula $x = \frac{-b}{2a}$ .

${ x = \frac{-4}{2(1)} \}$

${ x = 2 \}$

Finding the Y-Coordinate of the Vertex

Once we have found the x-coordinate of the vertex, we can find the y-coordinate of the vertex by substituting the x-coordinate into the equation of the quadratic function.

Example: Finding the Y-Coordinate of the Vertex

Let's consider the quadratic function $f(x) = x^2 + 4x + 5$ . We have already found the x-coordinate of the vertex, which is $x = 2$ . Now, we need to find the y-coordinate of the vertex by substituting $x = 2$ into the equation of the quadratic function.

${ f(2) = 2^2 - 4(2) - 5 \}$

${ f(2) = 4 - 8 - 5 \}$

${ f(2) = -9 \}$

Conclusion

In this step, we learned how to find the vertex of a quadratic function using the formula $x = \frac{-b}{2a}$ . We also learned how to find the y-coordinate of the vertex by substituting the x-coordinate into the equation of the quadratic function. The vertex of a quadratic function is the maximum or minimum point on the graph of the function, and it is denoted by the point $(h, k)$ , where $h$ is the x-coordinate of the vertex and $k$ is the y-coordinate of the vertex.

Step 3: Graphing Quadratic Functions

In the next step, we will learn how to graph quadratic functions using the vertex form of the equation. We will also learn how to identify the x-intercepts and y-intercepts of a quadratic function.

Step 3: Graphing Quadratic Functions

Introduction

In the previous step, we learned how to find the vertex of a quadratic function using the formula $x = \frac{-b}{2a}$ . Now, we will learn how to graph quadratic functions using the vertex form of the equation. The vertex form of the equation is $f(x) = a(x - h)^2 + k$ , where $(h, k)$ is the vertex of the function.

What is the Vertex Form of the Equation?

The vertex form of the equation is a way of writing the equation of a quadratic function in a form that makes it easy to graph the function. The vertex form of the equation is $f(x) = a(x - h)^2 + k$ , where $(h, k)$ is the vertex of the function. The vertex form of the equation is derived from the standard form of the equation by completing the square.

Graphing Quadratic Functions

To graph a quadratic function, we need to find the vertex of the function and the x-intercepts of the function. The vertex of the function is the point on the graph where the function changes from decreasing to increasing or from increasing to decreasing. The x-intercepts of the function are the points on the graph where the function intersects the x-axis.

Example: Graphing a Quadratic Function

Let's consider the quadratic function $f(x) = (x - 2)^2 - 9$ . To graph this function, we need to find the vertex of the function and the x-intercepts of the function.

Finding the Vertex of the Function

The vertex of the function is the point on the graph where the function changes from decreasing to increasing or from increasing to decreasing. To find the vertex of the function, we need to find the x-coordinate of the vertex and the y-coordinate of the vertex.

Finding the X-Coordinate of the Vertex

The x-coordinate of the vertex is the value of $x$ that makes the function equal to zero. To find the x-coordinate of the vertex, we need to solve the equation $(x - 2)^2 - 9 = 0$ .

Solving the Equation

To solve the equation $(x - 2)^2 - 9 = 0$ , we need to isolate the variable $x$ . We can do this by adding $9$ to both sides of the equation and then taking the square root of both sides of the equation.

Finding the Y-Coordinate of the Vertex

Once we have found the x-coordinate of the vertex, we can find the y-coordinate of the vertex by substituting the x-coordinate into the equation of the function.

Finding the X-Intercepts of the Function

The x-intercepts of the function are the points on the graph where the function intersects the x-axis. To find the x-intercepts of the function, we need to solve the equation $(x - 2)^2 - 9 = 0$ .

Solving the Equation

Graphing the Function

Once we have found the vertex of the function and the x-intercepts of the function, we can graph the function. We can do this by plotting the vertex and the x-intercepts on a coordinate plane and then drawing a smooth curve through the points.

Conclusion

In this step, we learned how to graph quadratic functions using the vertex form of the equation. We also learned how to find the x-intercepts and y-intercepts of a quadratic function. The vertex form of the equation is a way of writing the equation of a quadratic function in a form that makes it easy to graph the function.

Step 4: Identifying the X-Intercepts and Y-Intercepts of a Quadratic Function

Introduction

In the previous step, we learned how to graph quadratic functions using the vertex form of the equation. Now, we will learn how to identify the x-intercepts and y-intercepts of a quadratic function.

What are the X-Intercepts and Y-Intercepts of a Quadratic Function?

The x-intercepts of a quadratic function are the points on the graph where the function intersects the x-axis. The y-intercepts of a quadratic function are the points on the graph where the function intersects the y-axis.

Finding the X-Intercepts of a Quadratic Function

To find the x-intercepts of a quadratic function, we need to solve the equation $f(x) = 0$ . This equation is derived from the equation of the quadratic function by setting the function equal to zero.

Example: Finding the X-Intercepts of a Quadratic Function

Let's consider the quadratic function $f(x) = (x - 2)^2 - 9$ . To find the x-intercepts of this function, we need to solve the equation $(x - 2)^2 - 9 = 0$ .

Solving the Equation

Finding the Y-Intercepts of a Quadratic Function

To find the y-intercepts of a quadratic function, we need to find the value of the function at $x = 0$ . This value is the y-intercept of the function.

Example: Finding the Y-Intercepts of a Quadratic Function

Let's consider the quadratic function $f(x) = (x - 2)^2 - 9$ . To find the y-intercept of this function, we need to find the value of the function at $x = 0$ .

Finding the Value of the Function at x = 0

To find the value of the function at $x = 0$ , we need to substitute $x = 0$ into the equation of the function.

Conclusion

Introduction

In the previous step, we learned how to graph quadratic functions using the vertex form of the equation. Now, we will learn how to identify the x-intercepts and y-intercepts of a quadratic function.

Q&A: Identifying the X-Intercepts and Y-Intercepts of a Quadratic Function

Q: What are the x-intercepts of a quadratic function?

A: The x-intercepts of a quadratic function are the points on the graph where the function intersects the x-axis. These points are also known as the roots of the function.

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

A: To find the x-intercepts of a quadratic function, you need to solve the equation $f(x) = 0$ . This equation is derived from the equation of the quadratic function by setting the function equal to zero.

Q: What are the y-intercepts of a quadratic function?

A: The y-intercepts of a quadratic function are the points on the graph where the function intersects the y-axis. These points are also known as the y-values of the function.

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

A: To find the y-intercepts of a quadratic function, you need to find the value of the function at $x = 0$ . This value is the y-intercept of the function.

Q: Can a quadratic function have more than two x-intercepts?

A: No, a quadratic function can have at most two x-intercepts. This is because a quadratic function is a polynomial of degree two, and it can have at most two roots.

Q: Can a quadratic function have more than one y-intercept?

A: No, a quadratic function can have at most one y-intercept. This is because a quadratic function is a polynomial of degree two, and it can have at most one value at $x = 0$ .

Q: How do I graph a quadratic function if I know its x-intercepts and y-intercept?

A: To graph a quadratic function if you know its x-intercepts and y-intercept, you can use the following steps:

Plot the x-intercepts on a coordinate plane.
Plot the y-intercept on a coordinate plane.
Draw a smooth curve through the points.

Q: Can I use the x-intercepts and y-intercept to find the vertex of a quadratic function?

A: Yes, you can use the x-intercepts and y-intercept to find 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 from increasing to decreasing.

Q: How do I find the vertex of a quadratic function using the x-intercepts and y-intercept?

A: To find the vertex of a quadratic function using the x-intercepts and y-intercept, you can use the following steps:

Find the midpoint of the x-intercepts.
Find the value of the function at the midpoint.
The point $(h, k)$ is the vertex of the function.

Conclusion

In this step, we learned how to identify the x-intercepts and y-intercepts of a quadratic function. We also learned how to graph a quadratic function if we know its x-intercepts and y-intercept. Additionally, we learned how to find the vertex of a quadratic function using the x-intercepts and y-intercept.

Step 5: Solving Quadratic Equations

In the next step, we will learn how to solve quadratic equations using the quadratic formula. We will also learn how to use the quadratic formula to find the solutions to quadratic equations.

Step 5: Solving Quadratic Equations

Introduction

In the previous step, we learned how to identify the x-intercepts and y-intercepts of a quadratic function. Now, we will learn how to solve quadratic equations using the quadratic formula.

What is the Quadratic Formula?

The quadratic formula is a formula that is used to solve quadratic equations. It is given by:

${ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} }$

How Do I Use the Quadratic Formula to Solve a Quadratic Equation?

To use the quadratic formula to solve a quadratic equation, you need to follow these steps:

Write the quadratic equation in the form $ax^2 + bx + c = 0$ .
Identify the values of $a$ , $b$ , and $c$ .
Plug these values into the quadratic formula.
Simplify the expression.
Solve for $x$ .

Example: Solving a Quadratic Equation Using the Quadratic Formula

Let's consider the quadratic equation $x^2 + 4x + 4 = 0$ . To solve this equation using the quadratic formula, we need to follow these steps:

Write the quadratic equation in the form $ax^2 + bx + c = 0$ .
Identify the values of $a$ , $b$ , and $c$ .
Plug these values into the quadratic formula.
Simplify the expression.
Solve for $x$ .

Conclusion

In this step, we learned how to solve quadratic equations using the quadratic formula. We also learned how to use the quadratic formula to find the solutions to quadratic equations.

Step 6: Graphing Quadratic Functions

In the next step, we will learn how to graph quadratic functions using the vertex form of the equation. We will also learn how to identify the x-intercepts and y-intercepts of a quadratic function.

Step 6: Graphing Quadratic Functions

Introduction

In the previous step, we learned how to solve quadratic equations using the quadratic formula. Now, we will learn how to graph quadratic functions using the vertex form of the equation.

What is the Vertex Form of the Equation?

How Do I Graph a Quadratic Function Using the Vertex Form of the Equation?

To graph a quadratic function using the vertex form of the equation, you need to follow these steps:

Write the equation of the quadratic function in the vertex form.
Identify the values of $a$ , $h$ , and $k$ .
Plot the vertex on a coordinate plane.
Draw a smooth curve through the points.

Example: Graphing a Quadratic Function Using the Vertex Form of the Equation

Let's consider the quadratic function $f(x) = (x - 2)^2 - 9$ . To graph this function using the vertex form of the equation, we need to follow these steps:

Write the equation of the quadratic function in the vertex form.
Identify the values of $a$ , $h$ , and $k$ .
Plot the vertex on a coordinate plane.
Draw a smooth curve through the points.

Conclusion

In this step, we learned how to graph quadratic functions using the vertex form of the equation. We also learned how to identify the x-intercepts and y-intercepts of a quadratic function.