Graph The Function $f(x)=2x^2-4x-4$.1. Plot The Vertex Of The Parabola.2. Plot Another Point On The Parabola. Note: If You Make A Mistake, You Can Erase Your Parabola By Selecting The Second Point And Placing It On Top Of The First.

Feb 26, 2025 by ADMIN 235 views

**Graphing the Function $f(x)=2x^2-4x-4$**

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 $f(x)=2x^2-4x-4$ . We will first identify the vertex of the parabola and then plot another point on the parabola.

Step 1: Identify the Vertex of the Parabola

The vertex of a parabola is the point where the parabola changes direction. To find the vertex, 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 $b=-4$ . Plugging these values into the formula, we get:

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

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

$f(1)=2(1)^2-4(1)-4$ $f(1)=2-4-4$ $f(1)=-6$

So, the vertex of the parabola is at the point $(1,-6)$ .

Step 2: Plot the Vertex of the Parabola

To plot the vertex of the parabola, we can use a graphing tool or a piece of graph paper. If we are using a graphing tool, we can enter the coordinates of the vertex and plot the point. If we are using graph paper, we can draw a small dot at the point $(1,-6)$ .

Step 3: Plot Another Point on the Parabola

To plot another point on the parabola, we can choose a value of x and plug it into the function to find the corresponding value of y. Let's choose x=0 as our new x-coordinate. Plugging this value into the function, we get:

$f(0)=2(0)^2-4(0)-4$ $f(0)=-4$

So, the point (0,-4) is on the parabola.

Step 4: Plot the Point (0,-4)

To plot the point (0,-4), we can use a graphing tool or a piece of graph paper. If we are using a graphing tool, we can enter the coordinates of the point and plot it. If we are using graph paper, we can draw a small dot at the point (0,-4).

Conclusion

In this article, we graphed the function $f(x)=2x^2-4x-4$ . We identified the vertex of the parabola and plotted it on the graph. We also plotted another point on the parabola by choosing a value of x and plugging it into the function. By following these steps, we can graph any quadratic function and visualize its behavior.

Graphing Quadratic Functions

Graphing quadratic functions is an essential skill in mathematics, and it can be used to model various real-world situations. 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.

Properties of Quadratic Functions

Quadratic functions have several important properties that we need to understand in order to graph them. These properties include:

Vertex: The vertex of a parabola is the point where the parabola changes direction. It is the minimum or maximum point of the parabola.
Axis of Symmetry: The axis of symmetry of a parabola is a vertical line that passes through the vertex. It is the line of symmetry of the parabola.
Intercepts: The x-intercepts of a parabola are the points where the parabola intersects the x-axis. The y-intercept of a parabola is the point where the parabola intersects the y-axis.

Graphing Quadratic Functions

To graph a quadratic function, we need to identify the vertex, axis of symmetry, and intercepts of the parabola. We can use the following steps to graph a quadratic function:

Identify the Vertex: The vertex of a parabola is the point where the parabola changes direction. We can use the formula $x=-\frac{b}{2a}$ to find the x-coordinate of the vertex.
Identify the Axis of Symmetry: The axis of symmetry of a parabola is a vertical line that passes through the vertex. We can use the formula $x=-\frac{b}{2a}$ to find the equation of the axis of symmetry.
Identify the Intercepts: The x-intercepts of a parabola are the points where the parabola intersects the x-axis. We can use the formula $f(x)=0$ to find the x-intercepts. The y-intercept of a parabola is the point where the parabola intersects the y-axis. We can use the formula $f(0)=c$ to find the y-intercept.
Plot the Vertex: We can plot the vertex of the parabola by using a graphing tool or a piece of graph paper.
Plot the Axis of Symmetry: We can plot the axis of symmetry of the parabola by using a graphing tool or a piece of graph paper.
Plot the Intercepts: We can plot the intercepts of the parabola by using a graphing tool or a piece of graph paper.

Example

Let's graph the quadratic function $f(x)=x^2-4x-3$ . We can use the following steps to graph this function:

Identify the Vertex: The vertex of the parabola is the point where the parabola changes direction. We can use the formula $x=-\frac{b}{2a}$ to find the x-coordinate of the vertex.

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

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

$f(2)=(2)^2-4(2)-3$ $f(2)=4-8-3$ $f(2)=-7$

So, the vertex of the parabola is at the point $(2,-7)$ .

Identify the Axis of Symmetry: The axis of symmetry of the parabola is a vertical line that passes through the vertex. We can use the formula $x=-\frac{b}{2a}$ to find the equation of the axis of symmetry.

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

So, the equation of the axis of symmetry is $x=2$ .

Identify the Intercepts: The x-intercepts of the parabola are the points where the parabola intersects the x-axis. We can use the formula $f(x)=0$ to find the x-intercepts.

$f(x)=x^2-4x-3$ $f(x)=(x-3)(x+1)$

So, the x-intercepts of the parabola are at the points $(3,0)$ and $(-1,0)$ .

The y-intercept of the parabola is the point where the parabola intersects the y-axis. We can use the formula $f(0)=c$ to find the y-intercept.

$f(0)=(0)^2-4(0)-3$ $f(0)=-3$

So, the y-intercept of the parabola is at the point $(0,-3)$ .

Plot the Vertex: We can plot the vertex of the parabola by using a graphing tool or a piece of graph paper.
Plot the Axis of Symmetry: We can plot the axis of symmetry of the parabola by using a graphing tool or a piece of graph paper.
Plot the Intercepts: We can plot the intercepts of the parabola by using a graphing tool or a piece of graph paper.

Conclusion

Introduction

Graphing quadratic functions is an essential skill in mathematics, and it can be used to model various real-world situations. In this article, we will provide a Q&A section to help you better understand how to graph quadratic functions.

Q: What is a quadratic function?

A: A quadratic function is a polynomial function of degree two, which means it has a highest power of two. It is of the form $f(x)=ax^2+bx+c$ , where $a$ , $b$ , and $c$ are constants.

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the point where the parabola changes direction. It is the minimum or maximum point of the parabola.

Q: How do I find the vertex of a parabola?

A: To find the vertex of a parabola, you can use the formula $x=-\frac{b}{2a}$ , where $a$ and $b$ are the coefficients of the quadratic function. Once you have the x-coordinate of the vertex, you can find the y-coordinate by plugging the x-coordinate into the function.

Q: What is the axis of symmetry of a parabola?

A: The axis of symmetry of a parabola is a vertical line that passes through the vertex. It is the line of symmetry of the parabola.

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

A: To find the axis of symmetry of a parabola, you can use the formula $x=-\frac{b}{2a}$ , where $a$ and $b$ are the coefficients of the quadratic function. This will give you the equation of the axis of symmetry.

Q: What are the intercepts of a parabola?

A: The x-intercepts of a parabola are the points where the parabola intersects the x-axis. The y-intercept of a parabola is the point where the parabola intersects the y-axis.

Q: How do I find the intercepts of a parabola?

A: To find the x-intercepts of a parabola, you can set the function equal to zero and solve for x. To find the y-intercept of a parabola, you can plug x=0 into the function.

Q: How do I graph a quadratic function?

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

Identify the Vertex: Find the vertex of the parabola using the formula $x=-\frac{b}{2a}$ .
Identify the Axis of Symmetry: Find the axis of symmetry of the parabola using the formula $x=-\frac{b}{2a}$ .
Identify the Intercepts: Find the x-intercepts of the parabola by setting the function equal to zero and solving for x. Find the y-intercept of the parabola by plugging x=0 into the function.
Plot the Vertex: Plot the vertex of the parabola on the graph.
Plot the Axis of Symmetry: Plot the axis of symmetry of the parabola on the graph.
Plot the Intercepts: Plot the intercepts of the parabola on the graph.

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

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

Not identifying the vertex: Make sure to find the vertex of the parabola using the formula $x=-\frac{b}{2a}$ .
Not identifying the axis of symmetry: Make sure to find the axis of symmetry of the parabola using the formula $x=-\frac{b}{2a}$ .
Not identifying the intercepts: Make sure to find the x-intercepts of the parabola by setting the function equal to zero and solving for x. Make sure to find the y-intercept of the parabola by plugging x=0 into the function.
Not plotting the vertex, axis of symmetry, and intercepts: Make sure to plot the vertex, axis of symmetry, and intercepts of the parabola on the graph.

Conclusion

Graphing quadratic functions is an essential skill in mathematics, and it can be used to model various real-world situations. By following the steps outlined in this article, you can graph any quadratic function and visualize its behavior. Remember to identify the vertex, axis of symmetry, and intercepts of the parabola, and plot them on the graph.