Graph The Function F ( X ) = − X 2 + 6 X − 16 F(x) = -x^2 + 6x - 16 F ( X ) = − X 2 + 6 X − 16 .

Mar 8, 2025 by ADMIN 97 views

Graphing the Function $f(x) = -x^2 + 6x - 16$

Introduction

Graphing a function is a crucial aspect of mathematics, and it involves visualizing the relationship between the input and output values of the function. In this article, we will focus on graphing the function $f(x) = -x^2 + 6x - 16$ . This function is a quadratic function, and it represents a parabola on the coordinate plane. We will explore the properties of this function, including its vertex, axis of symmetry, and x-intercepts.

Understanding Quadratic Functions

Quadratic functions are of the form $f(x) = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants. The graph of a quadratic function is a parabola, which is a U-shaped curve. The parabola can open upwards or downwards, depending on the value of $a$ . If $a$ is positive, the parabola opens upwards, and if $a$ is negative, the parabola opens downwards.

In the case of the function $f(x) = -x^2 + 6x - 16$ , the coefficient of the $x^2$ term is $-1$ , which means that the parabola opens downwards. This is because the value of $a$ is negative.

Finding the Vertex

The vertex of a parabola is the highest or lowest point on the graph. It is the point where the parabola changes direction. To find the vertex of the function $f(x) = -x^2 + 6x - 16$ , we can use the formula $x = -\frac{b}{2a}$ . In this case, $a = -1$ and $b = 6$ , so we have:

$x = -\frac{6}{2(-1)}$ $x = -\frac{6}{-2}$ $x = 3$

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

$f(3) = -(3)^2 + 6(3) - 16$ $f(3) = -9 + 18 - 16$ $f(3) = -7$

Therefore, the vertex of the parabola is at the point $(3, -7)$ .

Finding the Axis of Symmetry

The axis of symmetry of a parabola is a vertical line that passes through the vertex. It is the line that divides the parabola into two equal parts. To find the equation of the axis of symmetry, we can use the formula $x = -\frac{b}{2a}$ . In this case, we already know that the x-coordinate of the vertex is $3$ , so the equation of the axis of symmetry is:

$x = 3$

Finding the X-Intercepts

The x-intercepts of a parabola are the points where the graph intersects the x-axis. To find the x-intercepts of the function $f(x) = -x^2 + 6x - 16$ , we can set the function equal to zero and solve for $x$ :

$-x^2 + 6x - 16 = 0$ $x^2 - 6x + 16 = 0$ $(x - 4)(x - 4) = 0$ $x - 4 = 0$ $x = 4$

Therefore, the x-intercept of the parabola is at the point $(4, 0)$ .

Graphing the Function

Now that we have found the vertex, axis of symmetry, and x-intercepts of the function $f(x) = -x^2 + 6x - 16$ , we can graph the function. The graph of the function is a parabola that opens downwards. The vertex of the parabola is at the point $(3, -7)$ , and the axis of symmetry is the line $x = 3$ . The x-intercept of the parabola is at the point $(4, 0)$ .

Conclusion

Graphing the function $f(x) = -x^2 + 6x - 16$ involves understanding the properties of quadratic functions, including the vertex, axis of symmetry, and x-intercepts. By using the formula $x = -\frac{b}{2a}$ , we can find the x-coordinate of the vertex, and by plugging this value into the function, we can find the y-coordinate. We can also find the equation of the axis of symmetry and the x-intercepts of the parabola. By graphing the function, we can visualize the relationship between the input and output values of the function.

Applications of Graphing Quadratic Functions

Graphing quadratic functions has many applications in mathematics and science. For example, it can be used to model the motion of objects under the influence of gravity, to describe the shape of a parabolic mirror, or to find the maximum or minimum value of a quadratic function.

Real-World Examples

Here are some real-world examples of graphing quadratic functions:

Projectile Motion: The trajectory of a projectile, such as a thrown ball or a rocket, can be modeled using a quadratic function. The graph of the function can be used to predict the maximum height and range of the projectile.
Optimization: Quadratic functions can be used to find the maximum or minimum value of a function. For example, a company may want to maximize its profit by finding the optimal price and quantity to sell a product.
Design: Quadratic functions can be used to design curves and shapes, such as the shape of a parabolic mirror or the curve of a road.

Final Thoughts

Introduction

In our previous article, we explored the properties of the function $f(x) = -x^2 + 6x - 16$ , including its vertex, axis of symmetry, and x-intercepts. We also discussed the importance of graphing quadratic functions in mathematics and science. In this article, we will answer some common questions related to graphing the function $f(x) = -x^2 + 6x - 16$ .

Q: What is the vertex of the parabola?

A: The vertex of the parabola is the highest or lowest point on the graph. To find the vertex, we can use the formula $x = -\frac{b}{2a}$ . In this case, $a = -1$ and $b = 6$ , so we have:

$x = -\frac{6}{2(-1)}$ $x = -\frac{6}{-2}$ $x = 3$

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

$f(3) = -(3)^2 + 6(3) - 16$ $f(3) = -9 + 18 - 16$ $f(3) = -7$

Therefore, the vertex of the parabola is at the point $(3, -7)$ .

Q: What is the axis of symmetry?

A: The axis of symmetry is a vertical line that passes through the vertex. It is the line that divides the parabola into two equal parts. To find the equation of the axis of symmetry, we can use the formula $x = -\frac{b}{2a}$ . In this case, we already know that the x-coordinate of the vertex is $3$ , so the equation of the axis of symmetry is:

$x = 3$

Q: What are the x-intercepts of the parabola?

A: The x-intercepts of the parabola are the points where the graph intersects the x-axis. To find the x-intercepts, we can set the function equal to zero and solve for $x$ :

$-x^2 + 6x - 16 = 0$ $x^2 - 6x + 16 = 0$ $(x - 4)(x - 4) = 0$ $x - 4 = 0$ $x = 4$

Therefore, the x-intercept of the parabola is at the point $(4, 0)$ .

Q: How do I graph the function?

A: To graph the function, you can use a graphing calculator or a computer program. You can also use a piece of graph paper and a pencil to draw the graph by hand. To draw the graph by hand, you can start by plotting the vertex and the x-intercepts. Then, you can use a ruler to draw a smooth curve through the points.

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

A: Graphing quadratic functions has many real-world applications, including:

Projectile Motion: The trajectory of a projectile, such as a thrown ball or a rocket, can be modeled using a quadratic function. The graph of the function can be used to predict the maximum height and range of the projectile.
Optimization: Quadratic functions can be used to find the maximum or minimum value of a function. For example, a company may want to maximize its profit by finding the optimal price and quantity to sell a product.
Design: Quadratic functions can be used to design curves and shapes, such as the shape of a parabolic mirror or the curve of a road.

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

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

Not using the correct formula: Make sure to use the correct formula to find the vertex, axis of symmetry, and x-intercepts.
Not plugging in the correct values: Make sure to plug in the correct values into the function to find the y-coordinate of the vertex.
Not drawing a smooth curve: Make sure to draw a smooth curve through the points to get an accurate graph.

Q: How can I use graphing quadratic functions in my everyday life?

A: Graphing quadratic functions can be used in many everyday situations, including:

Predicting the trajectory of a projectile: If you are playing a sport, such as basketball or soccer, you can use graphing quadratic functions to predict the trajectory of the ball.
Finding the maximum or minimum value of a function: If you are trying to maximize your profit or minimize your cost, you can use graphing quadratic functions to find the optimal solution.
Designing curves and shapes: If you are designing a curve or shape, such as a parabolic mirror or a road, you can use graphing quadratic functions to get an accurate design.