Graph The Function F ( X ) = 3 X 2 + 12 X − 15 F(x) = 3x^2 + 12x - 15 F ( X ) = 3 X 2 + 12 X − 15 .

Mar 8, 2025 by ADMIN 100 views

Graphing the Function $f(x) = 3x^2 + 12x - 15$

Introduction

Graphing a quadratic function is an essential skill in mathematics, and it can be used to model various real-world situations. In this article, we will focus on graphing the function $f(x) = 3x^2 + 12x - 15$ . We will start by understanding the properties of the function, and then we will use various methods to graph it.

Understanding the Function

The given function is a quadratic function in the form of $f(x) = ax^2 + bx + c$ , where $a = 3$ , $b = 12$ , and $c = -15$ . To understand the properties of the function, we need to find its vertex, axis of symmetry, and x-intercepts.

Finding the Vertex

The vertex of a quadratic function is the point where the function changes from decreasing to increasing or vice versa. It is given by the formula $x = -\frac{b}{2a}$ . In this case, $a = 3$ and $b = 12$ , so the x-coordinate of the vertex is $x = -\frac{12}{2(3)} = -2$ .

To find the y-coordinate of the vertex, we need to substitute the x-coordinate into the function. So, $f(-2) = 3(-2)^2 + 12(-2) - 15 = 12 - 24 - 15 = -27$ . Therefore, the vertex of the function is $(-2, -27)$ .

Finding the Axis of Symmetry

The axis of symmetry of a quadratic function is a vertical line that passes through the vertex. It is given by the equation $x = -\frac{b}{2a}$ . In this case, the axis of symmetry is $x = -2$ .

Finding the X-Intercepts

The x-intercepts of a quadratic function are the points where the function intersects the x-axis. They are given by the formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . In this case, $a = 3$ , $b = 12$ , and $c = -15$ , so the x-intercepts are $x = \frac{-12 \pm \sqrt{12^2 - 4(3)(-15)}}{2(3)} = \frac{-12 \pm \sqrt{144 + 180}}{6} = \frac{-12 \pm \sqrt{324}}{6} = \frac{-12 \pm 18}{6}$ .

Therefore, the x-intercepts are $x = \frac{-12 + 18}{6} = 1$ and $x = \frac{-12 - 18}{6} = -5$ .

Graphing the Function

Now that we have found the vertex, axis of symmetry, and x-intercepts, we can graph the function. We will use the following steps:

Plot the vertex at $(-2, -27)$ .
Plot the axis of symmetry at $x = -2$ .
Plot the x-intercepts at $(1, 0)$ and $(-5, 0)$ .
Use a ruler or a straightedge to draw a smooth curve through the points.

Here is the graph of the function:

Graph of the Function

Graph of the function

Conclusion

In this article, we have graphed the function $f(x) = 3x^2 + 12x - 15$ . We have found the vertex, axis of symmetry, and x-intercepts, and then we have used these points to graph the function. The graph shows that the function is a parabola that opens upward, with a vertex at $(-2, -27)$ and x-intercepts at $(1, 0)$ and $(-5, 0)$ .

Applications of Graphing Quadratic Functions

Graphing quadratic functions has many applications in real-world situations. Some of these applications include:

Modeling the trajectory of a projectile
Finding the maximum or minimum value of a function
Determining the stability of a system
Optimizing a process

Tips for Graphing Quadratic Functions

Here are some tips for graphing quadratic functions:

Make sure to find the vertex, axis of symmetry, and x-intercepts before graphing the function.
Use a ruler or a straightedge to draw a smooth curve through the points.
Check your graph for accuracy by plugging in some test points.
Use technology, such as a graphing calculator or a computer program, to graph the function if you are having trouble drawing it by hand.

Practice Problems

Here are some practice problems for graphing quadratic functions:

Graph the function $f(x) = 2x^2 - 5x + 3$ .
Find the vertex, axis of symmetry, and x-intercepts of the function $f(x) = x^2 + 4x - 5$ .
Graph the function $f(x) = -x^2 + 2x - 1$ .

Conclusion

Graphing quadratic functions is an essential skill in mathematics, and it has many applications in real-world situations. By following the steps outlined in this article, you can graph any quadratic function and find its vertex, axis of symmetry, and x-intercepts. Remember to check your graph for accuracy and to use technology if you are having trouble drawing it by hand.

Introduction

Graphing quadratic functions is an essential skill in mathematics, and it has many applications in real-world situations. In this article, we will answer some common questions about graphing quadratic functions.

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. It is given by the formula $x = -\frac{b}{2a}$ .

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

A: The axis of symmetry of a quadratic function is a vertical line that passes through the vertex. It is given by the equation $x = -\frac{b}{2a}$ .

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

A: The x-intercepts of a quadratic function are the points where the function intersects the x-axis. They are given by the formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ .

Q: How do I graph a quadratic function?

A: To graph a quadratic function, you need to find the vertex, axis of symmetry, and x-intercepts. Then, you can use a ruler or a straightedge to draw a smooth curve through the points.

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

A: Some common mistakes to avoid when graphing quadratic functions include:

Not finding the vertex, axis of symmetry, and x-intercepts before graphing the function.
Not using a ruler or a straightedge to draw a smooth curve through the points.
Not checking the graph for accuracy by plugging in some test points.

Q: Can I use technology to graph quadratic functions?

A: Yes, you can use technology, such as a graphing calculator or a computer program, to graph quadratic functions. This can be helpful if you are having trouble drawing the graph by hand.

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

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

Modeling the trajectory of a projectile
Finding the maximum or minimum value of a function
Determining the stability of a system
Optimizing a process

Q: How do I determine the direction of the parabola?

A: To determine the direction of the parabola, you need to look at the coefficient of the $x^2$ term. If the coefficient is positive, the parabola opens upward. If the coefficient is negative, the parabola opens downward.

Q: Can I graph a quadratic function that has no x-intercepts?

A: Yes, you can graph a quadratic function that has no x-intercepts. In this case, the graph will be a parabola that does not intersect the x-axis.

Q: How do I graph a quadratic function that has a complex x-intercept?

A: To graph a quadratic function that has a complex x-intercept, you need to use the formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . If the result is a complex number, you can use the fact that complex numbers come in conjugate pairs to find the other x-intercept.

Q: Can I use the graph of a quadratic function to find the maximum or minimum value of the function?

A: Yes, you can use the graph of a quadratic function to find the maximum or minimum value of the function. The maximum or minimum value will occur at the vertex of the parabola.

Q: How do I determine the stability of a system using the graph of a quadratic function?

A: To determine the stability of a system using the graph of a quadratic function, you need to look at the direction of the parabola. If the parabola opens upward, the system is stable. If the parabola opens downward, the system is unstable.

Q: Can I use the graph of a quadratic function to optimize a process?

A: Yes, you can use the graph of a quadratic function to optimize a process. The maximum or minimum value of the function will occur at the vertex of the parabola, and this can be used to optimize the process.