Graph The Function F ( X ) = − 2 X 2 + 12 X − 16 F(x) = -2x^2 + 12x - 16 F ( X ) = − 2 X 2 + 12 X − 16 . Plot Points To Show The Intercepts.Instructions:1. Use The Parabola Button And Click On The Coordinate Plane At The Vertex Location, Then Click On The Y Y Y -intercept Location.2. To Plot The

Introduction

Graphing quadratic functions is an essential skill in mathematics, particularly in algebra and calculus. Quadratic functions are polynomial functions of degree two, which means the highest power of the variable is two. In this article, we will focus on graphing the quadratic function $f(x) = -2x^2 + 12x - 16$. We will use the Desmos graphing calculator to visualize the graph and plot points to show the intercepts.

Understanding Quadratic Functions

A quadratic function can be written in the general form:

$f(x) = ax^2 + bx + c$

where $a$, $b$, and $c$ are constants, and $x$ is the variable. 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.

Graphing the Quadratic Function

To graph the quadratic function $f(x) = -2x^2 + 12x - 16$, we can use the Desmos graphing calculator. We will follow the instructions provided:

1. Use the Parabola button and click on the coordinate plane at the vertex location, then click on the $y$-intercept location.

To graph the parabola, we need to find the vertex location and the $y$-intercept location. The vertex location is the point on the parabola where the curve changes direction. The $y$-intercept location is the point on the parabola where the curve intersects the $y$-axis.

To find the vertex location, we can use the formula:

$x = -\frac{b}{2a}$

where $a$ and $b$ are the coefficients of the quadratic function. In this case, $a = -2$ and $b = 12$. Plugging in these values, we get:

$x = -\frac{12}{2(-2)} = 3$

So, the vertex location is $(3, f(3))$. To find the $y$-intercept location, we can plug in $x = 0$ into the quadratic function:

$f(0) = -2(0)^2 + 12(0) - 16 = -16$

So, the $y$-intercept location is $(0, -16)$.

Plotting Points to Show the Intercepts

To plot points to show the intercepts, we need to find the $x$-intercepts and the $y$-intercept. The $x$-intercepts are the points on the parabola where the curve intersects the $x$-axis. To find the $x$-intercepts, we can set $y = 0$ and solve for $x$:

$-2x^2 + 12x - 16 = 0$

We can solve this quadratic equation using the quadratic formula:

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

where $a = -2$, $b = 12$, and $c = -16$. Plugging in these values, we get:

$x = \frac{-12 \pm \sqrt{12^2 - 4(-2)(-16)}}{2(-2)}$

$x = \frac{-12 \pm \sqrt{144 - 128}}{-4}$

$x = \frac{-12 \pm \sqrt{16}}{-4}$

$x = \frac{-12 \pm 4}{-4}$

So, the $x$-intercepts are $x = 2$ and $x = 4$.

Conclusion

In this article, we graphed the quadratic function $f(x) = -2x^2 + 12x - 16$ using the Desmos graphing calculator. We found the vertex location and the $y$-intercept location, and plotted points to show the intercepts. We also discussed the importance of graphing quadratic functions and the different types of quadratic functions.

Graph

f(x) = -2x^2 + 12x - 16

Vertex Location

The vertex location is $(3, f(3))$. To find the $y$-coordinate, we can plug in $x = 3$ into the quadratic function:

$f(3) = -2(3)^2 + 12(3) - 16$

$f(3) = -2(9) + 36 - 16$

$f(3) = -18 + 36 - 16$

$f(3) = 2$

So, the vertex location is $(3, 2)$.

y-Intercept Location

The $y$-intercept location is $(0, -16)$.

x-Intercepts

The $x$-intercepts are $x = 2$ and $x = 4$.

Table of Values

$x$	$f(x)$
0	-16
2	0
3	2
4	0
5	-2

Discussion

Graphing quadratic functions is an essential skill in mathematics, particularly in algebra and calculus. Quadratic functions are polynomial functions of degree two, which means the highest power of the variable is two. In this article, we graphed the quadratic function $f(x) = -2x^2 + 12x - 16$ using the Desmos graphing calculator. We found the vertex location and the $y$-intercept location, and plotted points to show the intercepts. We also discussed the importance of graphing quadratic functions and the different types of quadratic functions.

References

• [1] "Graphing Quadratic Functions" by Math Open Reference
• [2] "Quadratic Functions" by Khan Academy
• [3] "Graphing Quadratic Functions" by Purplemath

Keywords

• Graphing quadratic functions
• Quadratic functions
• Parabola
• Vertex location
• y-intercept location
• x-intercepts
• Table of values
• Desmos graphing calculator
• Algebra
• Calculus
  Graphing Quadratic Functions: A Q&A Guide =====================================================

Introduction

Graphing quadratic functions is an essential skill in mathematics, particularly in algebra and calculus. Quadratic functions are polynomial functions of degree two, which means the highest power of the variable is two. In this article, we will provide a Q&A guide to help you understand and graph quadratic functions.

Q: What is a quadratic function?

A: A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. It can be written in the general form:

$f(x) = ax^2 + bx + c$

where $a$, $b$, and $c$ are constants, and $x$ is the variable.

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

A: The vertex location of a quadratic function is the point on the parabola where the curve changes direction. It can be found using the formula:

$x = -\frac{b}{2a}$

where $a$ and $b$ are the coefficients of the quadratic function.

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

A: To find the y-intercept location of a quadratic function, you can plug in $x = 0$ into the quadratic function:

$f(0) = a(0)^2 + b(0) + c$

$f(0) = c$

So, the y-intercept location is $(0, c)$.

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 $y = 0$ and solve for $x$:

$ax^2 + bx + c = 0$

You can solve this quadratic equation using the quadratic formula:

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

Q: What is the difference between a parabola that opens upwards and one that opens downwards?

A: A parabola that opens upwards has a positive leading coefficient ($a > 0$), while a parabola that opens downwards has a negative leading coefficient ($a < 0$).

Q: How do I graph a quadratic function using the Desmos graphing calculator?

A: To graph a quadratic function using the Desmos graphing calculator, you can follow these steps:

1. Enter the quadratic function into the calculator.
2. Use the Parabola button and click on the coordinate plane at the vertex location.
3. Click on the y-intercept location.
4. Plot points to show the intercepts.

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 to find the vertex location.
• Not plugging in the correct values into the quadratic function.
• Not using the correct method to find the x-intercepts.
• Not checking the work for errors.

Q: How do I check my work when graphing quadratic functions?

A: To check your work when graphing quadratic functions, you can:

• Use a calculator to check the vertex location and the y-intercept location.
• Use a graphing calculator to check the graph of the quadratic function.
• Check the work for errors by plugging in the correct values into the quadratic function.

Conclusion

Graphing quadratic functions is an essential skill in mathematics, particularly in algebra and calculus. Quadratic functions are polynomial functions of degree two, which means the highest power of the variable is two. In this article, we provided a Q&A guide to help you understand and graph quadratic functions. We also discussed some common mistakes to avoid and how to check your work when graphing quadratic functions.

Graph

f(x) = -2x^2 + 12x - 16

Vertex Location

The vertex location is $(3, f(3))$. To find the y-coordinate, we can plug in $x = 3$ into the quadratic function:

$f(3) = -2(3)^2 + 12(3) - 16$

$f(3) = -2(9) + 36 - 16$

$f(3) = -18 + 36 - 16$

$f(3) = 2$

So, the vertex location is $(3, 2)$.

y-Intercept Location

The y-intercept location is $(0, -16)$.

x-Intercepts

The x-intercepts are $x = 2$ and $x = 4$.

Table of Values

$x$	$f(x)$
0	-16
2	0
3	2
4	0
5	-2

Discussion

Graphing quadratic functions is an essential skill in mathematics, particularly in algebra and calculus. Quadratic functions are polynomial functions of degree two, which means the highest power of the variable is two. In this article, we provided a Q&A guide to help you understand and graph quadratic functions. We also discussed some common mistakes to avoid and how to check your work when graphing quadratic functions.

References

• [1] "Graphing Quadratic Functions" by Math Open Reference
• [2] "Quadratic Functions" by Khan Academy
• [3] "Graphing Quadratic Functions" by Purplemath

Keywords

• Graphing quadratic functions
• Quadratic functions
• Parabola
• Vertex location
• y-intercept location
• x-intercepts
• Table of values
• Desmos graphing calculator
• Algebra
• Calculus