Graphing the Function with a Scale

Oct 13, 2025 by magicalceylontours.com 76 views

Understanding the Function

The given function is a quadratic function in the form of g(x) = 2(x + 2)(x + 6). This function can be expanded to the form g(x) = 2(x^2 + 8x + 12). To graph this function, we need to identify its key features, such as its x-intercepts, y-intercept, and vertex.

Identifying Key Features

To identify the x-intercepts, we need to set the function equal to zero and solve for x. This gives us the equation 2(x^2 + 8x + 12) = 0. Dividing both sides by 2, we get x^2 + 8x + 12 = 0. This quadratic equation can be factored as (x + 2)(x + 6) = 0. Setting each factor equal to zero, we get x + 2 = 0 and x + 6 = 0. Solving for x, we get x = -2 and x = -6.

Finding the Y-Intercept

To find the y-intercept, we need to substitute x = 0 into the function. This gives us g(0) = 2(0^2 + 8(0) + 12) = 2(12) = 24. Therefore, the y-intercept is (0, 24).

Identifying the Vertex

The vertex of a quadratic function in the form of g(x) = ax^2 + bx + c is given by the formula x = -b/2a. In this case, a = 2 and b = 8. Plugging these values into the formula, we get x = -8/(2(2)) = -8/4 = -2. To find the y-coordinate of the vertex, we substitute x = -2 into the function. This gives us g(-2) = 2((-2)^2 + 8(-2) + 12) = 2(4 - 16 + 12) = 2(0) = 0. Therefore, the vertex is (-2, 0).

Graphing the Function

Now that we have identified the key features of the function, we can graph it. The graph of the function will be a parabola that opens upward, since the coefficient of the x^2 term is positive. The x-intercepts are (-2, 0) and (-6, 0), and the y-intercept is (0, 24). The vertex is (-2, 0).

Plotting Key Points

To plot key points on the graph, we need to substitute x-values into the function and find the corresponding y-values. Some key points to plot are:

(-2, 0)
(-6, 0)
(0, 24)
(-4, 16)
(-3, 20)

Graphing the Function with a Scale

To graph the function with an appropriate scale for both the x-axis and y-axis, we need to determine the range of values for both axes. The x-axis should range from -7 to 1, and the y-axis should range from 0 to 30.

Final Graph

Here is the final graph of the function g(x) = 2(x + 2)(x + 6):

Graph:

x	y
-7	0
-6	0
-5	4
-4	16
-3	20
-2	0
-1	4
0	24
1	40

Note: The graph is a parabola that opens upward, with x-intercepts at (-2, 0) and (-6, 0), and a y-intercept at (0, 24). The vertex is (-2, 0).

Conclusion

In this article, we graphed the function g(x) = 2(x + 2)(x + 6) and identified its key features, such as its x-intercepts, y-intercept, and vertex. We also plotted key points on the graph and determined the range of values for both the x-axis and y-axis. The final graph shows a parabola that opens upward, with x-intercepts at (-2, 0) and (-6, 0), and a y-intercept at (0, 24). The vertex is (-2, 0).

Key Takeaways

The function g(x) = 2(x + 2)(x + 6) is a quadratic function in the form of g(x) = ax^2 + bx + c.
The x-intercepts are (-2, 0) and (-6, 0).
The y-intercept is (0, 24).
The vertex is (-2, 0).
The graph of the function is a parabola that opens upward.

Real-World Applications

The graph of a quadratic function has many real-world applications, such as:

Modeling the trajectory of a projectile
Finding the maximum or minimum value of a function
Determining the x-intercepts and y-intercept of a function

Future Directions

In the future, we can explore other types of functions, such as polynomial functions and rational functions. We can also investigate the properties of functions, such as symmetry and periodicity.

References

[1] "Graphing Quadratic Functions" by Math Open Reference
[2] "Quadratic Functions" by Khan Academy
[3] "Graphing Functions" by Wolfram Alpha

Understanding the Function

Q&A

Q: What is the x-intercept of the function g(x) = 2(x + 2)(x + 6)?

A: The x-intercepts of the function g(x) = 2(x + 2)(x + 6) are (-2, 0) and (-6, 0).

Q: What is the y-intercept of the function g(x) = 2(x + 2)(x + 6)?

A: The y-intercept of the function g(x) = 2(x + 2)(x + 6) is (0, 24).

Q: What is the vertex of the function g(x) = 2(x + 2)(x + 6)?

A: The vertex of the function g(x) = 2(x + 2)(x + 6) is (-2, 0).

Q: How do I graph the function g(x) = 2(x + 2)(x + 6)?

A: To graph the function g(x) = 2(x + 2)(x + 6), you need to identify its key features, such as its x-intercepts, y-intercept, and vertex. You can then plot these points on a graph and draw a smooth curve through them.

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, and determining the x-intercepts and y-intercept of a function.

Q: Can I use graphing software to graph the function g(x) = 2(x + 2)(x + 6)?

A: Yes, you can use graphing software, such as Desmos or GeoGebra, to graph the function g(x) = 2(x + 2)(x + 6).

Q: How do I determine the range of values for both the x-axis and y-axis when graphing the function g(x) = 2(x + 2)(x + 6)?

A: To determine the range of values for both the x-axis and y-axis when graphing the function g(x) = 2(x + 2)(x + 6), you need to identify the x-intercepts and y-intercept of the function. You can then use these points to determine the range of values for both axes.