Graph The Function $f(x)=6(x-3)^2-6$.1. Plot The Vertex.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.

Graphing the Function $f(x)=6(x-3)^2-6$

Understanding the Function

The given function is a quadratic function in the form of $f(x)=a(x-h)^2+k$, where $(h,k)$ represents the vertex of the parabola. In this case, the function is $f(x)=6(x-3)^2-6$. To graph this function, we need to identify the vertex and another point on the parabola.

1. Plotting the Vertex

The vertex of the parabola is given by the values of $h$ and $k$ in the function. In this case, $h=3$ and $k=-6$. To plot the vertex, we need to locate the point $(3,-6)$ on the coordinate plane.

Vertex Form of a Quadratic Function

The vertex form of a quadratic function is given by $f(x)=a(x-h)^2+k$. This form is useful for graphing quadratic functions because it allows us to identify the vertex of the parabola directly from the function.

2. Plotting Another Point on the Parabola

To plot another point on the parabola, we need to choose a value of $x$ and substitute it into the function to find the corresponding value of $y$. Let's choose $x=0$ as our test point.

Finding the Value of $y$

To find the value of $y$, we need to substitute $x=0$ into the function:

$f(0)=6(0-3)^2-6$

$f(0)=6(-3)^2-6$

$f(0)=6(9)-6$

$f(0)=54-6$

$f(0)=48$

Plotting the Point

Now that we have the value of $y$, we can plot the point $(0,48)$ on the coordinate plane.

Graphing the Parabola

To graph the parabola, we need to connect the vertex and the other point we plotted. Since the vertex is at $(3,-6)$ and the other point is at $(0,48)$, we can draw a smooth curve through these two points to represent the parabola.

Properties of the Parabola

The parabola has a number of important properties that we can identify from the function. The vertex of the parabola is at $(3,-6)$, which means that the parabola opens upward. The axis of symmetry of the parabola is the vertical line $x=3$. The parabola has a minimum value of $-6$ at the vertex.

Graphing the Function

To graph the function, we can use a graphing calculator or a computer program to plot the parabola. We can also use a piece of graph paper and a pencil to draw the parabola by hand.

Conclusion

In this article, we graphed the function $f(x)=6(x-3)^2-6$ by identifying the vertex and another point on the parabola. We used the vertex form of a quadratic function to identify the vertex and plotted the point $(0,48)$ on the coordinate plane. We then connected the vertex and the other point to represent the parabola. We also identified the properties of the parabola, including the axis of symmetry and the minimum value.

Properties of the Parabola

• Vertex: The vertex of the parabola is at $(3,-6)$.
• Axis of Symmetry: The axis of symmetry of the parabola is the vertical line $x=3$.
• Minimum Value: The parabola has a minimum value of $-6$ at the vertex.

Graphing the Function

To graph the function, we can use a graphing calculator or a computer program to plot the parabola. We can also use a piece of graph paper and a pencil to draw the parabola by hand.

Real-World Applications

The graph of a quadratic function has many real-world applications. For example, the graph of a quadratic function can be used to model the trajectory of a projectile, the motion of a pendulum, or the growth of a population.

Conclusion

In this article, we graphed the function $f(x)=6(x-3)^2-6$ by identifying the vertex and another point on the parabola. We used the vertex form of a quadratic function to identify the vertex and plotted the point $(0,48)$ on the coordinate plane. We then connected the vertex and the other point to represent the parabola. We also identified the properties of the parabola, including the axis of symmetry and the minimum value.

Properties of the Parabola

• Vertex: The vertex of the parabola is at $(3,-6)$.
• Axis of Symmetry: The axis of symmetry of the parabola is the vertical line $x=3$.
• Minimum Value: The parabola has a minimum value of $-6$ at the vertex.

Graphing the Function

To graph the function, we can use a graphing calculator or a computer program to plot the parabola. We can also use a piece of graph paper and a pencil to draw the parabola by hand.

Real-World Applications

The graph of a quadratic function has many real-world applications. For example, the graph of a quadratic function can be used to model the trajectory of a projectile, the motion of a pendulum, or the growth of a population.

Final Thoughts

Graphing a quadratic function is an important skill in mathematics and has many real-world applications. By identifying the vertex and another point on the parabola, we can graph the function and identify its properties. We can use this skill to model real-world situations and make predictions about the behavior of a system.

References

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

Additional Resources

• [1] "Graphing Quadratic Functions" by Mathway
• [2] "Quadratic Functions" by Wolfram Alpha
• [3] "Graphing Quadratic Functions" by GeoGebra