Graph The Function And Provide The Attributes.${ G(x) = (x+6)^3 - 2 }$ { \begin{tabular}{|c|c|} \hline X$ & G ( X ) G(x) G ( X ) \ \hline & \ \hline & \ \hline & \ \hline & \ \hline & \ \hline & \ \hline \end{tabular} }$-

Introduction

Graphing a function is a crucial aspect of mathematics, as it allows us to visualize the behavior of the function and identify its key attributes. In this article, we will focus on graphing the function $g(x) = (x+6)^3 - 2$ and identifying its attributes.

Understanding the Function

The given function is $g(x) = (x+6)^3 - 2$. This is a cubic function, which means it has a degree of 3. The function is in the form of a polynomial, where the variable $x$ is raised to the power of 3.

Graphing the Function

To graph the function, we need to find the values of $x$ and $g(x)$ for different values of $x$. We can do this by plugging in different values of $x$ into the function and calculating the corresponding values of $g(x)$.

$x$	$g(x)$
-10	?
-8	?
-6	?
-4	?
-2	?
0	?
2	?
4	?
6	?
8	?
10	?

Let's calculate the values of $g(x)$ for each of these values of $x$.

Calculating $g(x)$ for $x = -10$

$g(-10) = ((-10)+6)^3 - 2$ $g(-10) = (-4)^3 - 2$ $g(-10) = -64 - 2$ $g(-10) = -66$

Calculating $g(x)$ for $x = -8$

$g(-8) = ((-8)+6)^3 - 2$ $g(-8) = (-2)^3 - 2$ $g(-8) = -8 - 2$ $g(-8) = -10$

Calculating $g(x)$ for $x = -6$

$g(-6) = ((-6)+6)^3 - 2$ $g(-6) = (0)^3 - 2$ $g(-6) = -2$

Calculating $g(x)$ for $x = -4$

$g(-4) = ((-4)+6)^3 - 2$ $g(-4) = (2)^3 - 2$ $g(-4) = 8 - 2$ $g(-4) = 6$

Calculating $g(x)$ for $x = -2$

$g(-2) = ((-2)+6)^3 - 2$ $g(-2) = (4)^3 - 2$ $g(-2) = 64 - 2$ $g(-2) = 62$

Calculating $g(x)$ for $x = 0$

$g(0) = ((0)+6)^3 - 2$ $g(0) = (6)^3 - 2$ $g(0) = 216 - 2$ $g(0) = 214$

Calculating $g(x)$ for $x = 2$

$g(2) = ((2)+6)^3 - 2$ $g(2) = (8)^3 - 2$ $g(2) = 512 - 2$ $g(2) = 510$

Calculating $g(x)$ for $x = 4$

$g(4) = ((4)+6)^3 - 2$ $g(4) = (10)^3 - 2$ $g(4) = 1000 - 2$ $g(4) = 998$

Calculating $g(x)$ for $x = 6$

$g(6) = ((6)+6)^3 - 2$ $g(6) = (12)^3 - 2$ $g(6) = 1728 - 2$ $g(6) = 1726$

Calculating $g(x)$ for $x = 8$

$g(8) = ((8)+6)^3 - 2$ $g(8) = (14)^3 - 2$ $g(8) = 2744 - 2$ $g(8) = 2742$

Calculating $g(x)$ for $x = 10$

$g(10) = ((10)+6)^3 - 2$ $g(10) = (16)^3 - 2$ $g(10) = 4096 - 2$ $g(10) = 4094$

Now that we have calculated the values of $g(x)$ for each of these values of $x$, we can plot the points on a graph.

Plotting the Graph

The graph of the function $g(x) = (x+6)^3 - 2$ is a cubic graph that opens upwards. The graph passes through the points $(-10, -66)$, $(-8, -10)$, $(-6, -2)$, $(-4, 6)$, $(-2, 62)$, $(0, 214)$, $(2, 510)$, $(4, 998)$, $(6, 1726)$, $(8, 2742)$, and $(10, 4094)$.

Identifying Attributes

Now that we have graphed the function, we can identify its key attributes.

Domain

The domain of the function is all real numbers, which means that the function is defined for all values of $x$.

Range

The range of the function is all real numbers greater than or equal to -2, which means that the function takes on all values greater than or equal to -2.

X-Intercept

The x-intercept of the function is the value of $x$ where the graph intersects the x-axis. In this case, the x-intercept is $x = -6$.

Y-Intercept

The y-intercept of the function is the value of $y$ where the graph intersects the y-axis. In this case, the y-intercept is $y = -2$.

Vertex

The vertex of the function is the point on the graph where the function changes from decreasing to increasing. In this case, the vertex is the point $(-6, -2)$.

Axis of Symmetry

The axis of symmetry of the function is the vertical line that passes through the vertex. In this case, the axis of symmetry is the line $x = -6$.

End Behavior

The end behavior of the function is the behavior of the function as $x$ approaches positive or negative infinity. In this case, the end behavior is that the function approaches positive infinity as $x$ approaches positive infinity and negative infinity as $x$ approaches negative infinity.

Conclusion

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Introduction

In our previous article, we graphed the function $g(x) = (x+6)^3 - 2$ and identified its key attributes. In this article, we will answer some common questions related to graphing and identifying attributes of functions.

Q: What is the domain of the function?

A: The domain of the function is all real numbers, which means that the function is defined for all values of $x$.

Q: What is the range of the function?

A: The range of the function is all real numbers greater than or equal to -2, which means that the function takes on all values greater than or equal to -2.

Q: What is the x-intercept of the function?

A: The x-intercept of the function is the value of $x$ where the graph intersects the x-axis. In this case, the x-intercept is $x = -6$.

Q: What is the y-intercept of the function?

A: The y-intercept of the function is the value of $y$ where the graph intersects the y-axis. In this case, the y-intercept is $y = -2$.

Q: What is the vertex of the function?

A: The vertex of the function is the point on the graph where the function changes from decreasing to increasing. In this case, the vertex is the point $(-6, -2)$.

Q: What is the axis of symmetry of the function?

A: The axis of symmetry of the function is the vertical line that passes through the vertex. In this case, the axis of symmetry is the line $x = -6$.

Q: What is the end behavior of the function?

A: The end behavior of the function is the behavior of the function as $x$ approaches positive or negative infinity. In this case, the end behavior is that the function approaches positive infinity as $x$ approaches positive infinity and negative infinity as $x$ approaches negative infinity.

Q: How do I graph a function?

A: To graph a function, you need to find the values of $x$ and $y$ for different values of $x$. You can do this by plugging in different values of $x$ into the function and calculating the corresponding values of $y$. Then, you can plot the points on a graph.

Q: What are some common types of functions?

A: Some common types of functions include:

• Linear functions: $f(x) = mx + b$
• Quadratic functions: $f(x) = ax^2 + bx + c$
• Cubic functions: $f(x) = ax^3 + bx^2 + cx + d$
• Polynomial functions: $f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$

Q: How do I identify the attributes of a function?

A: To identify the attributes of a function, you need to graph the function and look for the following:

• Domain: the set of all possible input values
• Range: the set of all possible output values
• X-intercept: the value of $x$ where the graph intersects the x-axis
• Y-intercept: the value of $y$ where the graph intersects the y-axis
• Vertex: the point on the graph where the function changes from decreasing to increasing
• Axis of symmetry: the vertical line that passes through the vertex
• End behavior: the behavior of the function as $x$ approaches positive or negative infinity

Conclusion

In this article, we answered some common questions related to graphing and identifying attributes of functions. We hope that this article has been helpful in understanding the concepts of graphing and identifying attributes of functions.