Graph This Function: $f(x) = X^2 - 4x - 5$Step 1: Identify $a$ And $b$. $a = \checkmark, \quad B = \checkmark$

Feb 28, 2025 by ADMIN 112 views

**Graphing Quadratic Functions: A Step-by-Step Guide**

Understanding Quadratic Functions

Quadratic functions are a type of polynomial function that can be written in the form $f(x) = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants. These functions have a parabolic shape and can be graphed using various methods. In this article, we will focus on graphing the quadratic function $f(x) = x^2 - 4x - 5$ .

Step 1: Identify $a$ and $b$

To graph a quadratic function, we need to identify the values of $a$ and $b$ in the function. In the given function $f(x) = x^2 - 4x - 5$ , we can see that $a = 1$ and $b = -4$ .

Why is it important to identify $a$ and $b$ ?

Identifying $a$ and $b$ is crucial in graphing a quadratic function because it helps us determine the direction and width of the parabola. If $a$ is positive, the parabola opens upwards, and if $a$ is negative, the parabola opens downwards. Similarly, if $b$ is positive, the parabola is wider, and if $b$ is negative, the parabola is narrower.

Step 2: Determine the Vertex

The vertex of a quadratic function is the point at which the parabola changes direction. To find the vertex, we can use the formula $x = -\frac{b}{2a}$ . Plugging in the values of $a$ and $b$ from the given function, we get:

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

Now that we have the x-coordinate of the vertex, we can find the y-coordinate by plugging the value of $x$ into the function:

$f(2) = (2)^2 - 4(2) - 5 = 4 - 8 - 5 = -9$

So, the vertex of the parabola is at the point $(2, -9)$ .

Why is the vertex important?

The vertex is an important point on the parabola because it helps us determine the direction and width of the parabola. If the vertex is at the origin $(0, 0)$ , the parabola is symmetrical about the x-axis. If the vertex is not at the origin, the parabola is not symmetrical about the x-axis.

Step 3: Find the x-Intercepts

The x-intercepts of a quadratic function are the points at which the parabola intersects the x-axis. To find the x-intercepts, we can set the function equal to zero and solve for $x$ :

$x^2 - 4x - 5 = 0$

We can factor the quadratic expression as:

$(x - 5)(x + 1) = 0$

Setting each factor equal to zero, we get:

$x - 5 = 0 \quad \text{or} \quad x + 1 = 0$

Solving for $x$ , we get:

$x = 5 \quad \text{or} \quad x = -1$

So, the x-intercepts of the parabola are at the points $(5, 0)$ and $(-1, 0)$ .

Why are the x-intercepts important?

The x-intercepts are important points on the parabola because they help us determine the width and direction of the parabola. If the x-intercepts are far apart, the parabola is wider. If the x-intercepts are close together, the parabola is narrower.

Step 4: Graph the Parabola

Now that we have identified the vertex and x-intercepts, we can graph the parabola. To do this, we can plot the vertex and x-intercepts on a coordinate plane and draw a smooth curve through the points.

How to graph a parabola?

To graph a parabola, we can follow these steps:

Plot the vertex on the coordinate plane.
Plot the x-intercepts on the coordinate plane.
Draw a smooth curve through the points, making sure to include the vertex and x-intercepts.

Conclusion

Graphing a quadratic function involves identifying the values of $a$ and $b$ , determining the vertex, finding the x-intercepts, and graphing the parabola. By following these steps, we can create a graph of the quadratic function $f(x) = x^2 - 4x - 5$ .

What are the benefits of graphing a quadratic function?

Graphing a quadratic function has several benefits, including:

Understanding the shape of the parabola: Graphing a quadratic function helps us understand the shape of the parabola, including its direction, width, and vertex.
Identifying key points: Graphing a quadratic function helps us identify key points on the parabola, including the vertex and x-intercepts.
Solving problems: Graphing a quadratic function can help us solve problems involving quadratic functions, such as finding the maximum or minimum value of the function.

What are some common applications of graphing quadratic functions?

Graphing quadratic functions has several common applications, including:

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

What are some common mistakes to avoid when graphing quadratic functions?

When graphing quadratic functions, there are several common mistakes to avoid, including:

Incorrectly identifying the vertex: Make sure to use the correct formula to find the vertex, and double-check your calculations.
Incorrectly finding the x-intercepts: Make sure to set the function equal to zero and solve for $x$ correctly.
Not including the vertex and x-intercepts: Make sure to include the vertex and x-intercepts on the graph, as these are critical points on the parabola.
Graphing Quadratic Functions: A Q&A Guide =============================================

Frequently Asked Questions

Q: What is the vertex of a quadratic function?

A: The vertex of a quadratic function is the point at which the parabola changes direction. It is the minimum or maximum point on the parabola, 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 can use the formula $x = -\frac{b}{2a}$ . This will give you the x-coordinate of the vertex. To find the y-coordinate, plug the value of $x$ into the function.

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

A: The x-intercepts of a quadratic function are the points at which the parabola intersects the x-axis. They are the solutions to the equation $f(x) = 0$ .

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

A: To find the x-intercepts of a quadratic function, you can set the function equal to zero and solve for $x$ . This will give you the x-coordinates of the x-intercepts.

Q: What is the significance of the vertex and x-intercepts in graphing a quadratic function?

A: The vertex and x-intercepts are critical points on the parabola that help determine the direction and width of the parabola. They are essential in graphing a quadratic function.

Q: How do I graph a quadratic function?

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

Plot the vertex on the coordinate plane.
Plot the x-intercepts on the coordinate plane.
Draw a smooth curve through the points, making sure to include the vertex and x-intercepts.

Q: What are some common applications of graphing quadratic functions?

A: Graphing quadratic functions has several common applications, including:

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

Q: What are some common mistakes to avoid when graphing quadratic functions?

A: When graphing quadratic functions, there are several common mistakes to avoid, including:

Incorrectly identifying the vertex: Make sure to use the correct formula to find the vertex, and double-check your calculations.
Incorrectly finding the x-intercepts: Make sure to set the function equal to zero and solve for $x$ correctly.
Not including the vertex and x-intercepts: Make sure to include the vertex and x-intercepts on the graph, as these are critical points on the parabola.

Q: How do I determine the direction of a quadratic function?

A: To determine the direction of a quadratic function, you can look at the value of $a$ . If $a$ is positive, the parabola opens upwards. If $a$ is negative, the parabola opens downwards.

Q: How do I determine the width of a quadratic function?

A: To determine the width of a quadratic function, you can look at the value of $b$ . If $b$ is positive, the parabola is wider. If $b$ is negative, the parabola is narrower.

Q: What are some real-world examples of graphing quadratic functions?

A: Some real-world examples of graphing quadratic functions include:

Designing a roller coaster: Graphing a quadratic function can help designers create a roller coaster with a smooth and thrilling ride.
Modeling population growth: Graphing a quadratic function can help model the growth of a population over time.
Optimizing a system: Graphing a quadratic function can help optimize a system, such as a bridge or a building.

Q: What are some tips for graphing quadratic functions?

A: Some tips for graphing quadratic functions include:

Use a graphing calculator: A graphing calculator can help you visualize the graph of a quadratic function.
Plot the vertex and x-intercepts: Make sure to include the vertex and x-intercepts on the graph, as these are critical points on the parabola.
Draw a smooth curve: Make sure to draw a smooth curve through the points, making sure to include the vertex and x-intercepts.