Problem 16 - Algebra and Trigonometry

Question

Sketch the graph of the function by first making a table of values. \(g(x)=(x+2)^{3}\)

Step-by-Step Solution

Answer

Plot the points based on \( g(x) = (x+2)^3 \) values and draw the cubic curve.

Step 1: Choose Values for x

To sketch the graph, we need to create a table of values for the function. Let's choose a range of x-values around the point where the function changes significantly. Typically, we choose values around the point of interest or around 0, so let's pick: \( x = -4, -3, -2, -1, 0, 1, 2 \).

Step 2: Calculate Corresponding y-values

Now, let's calculate the value of \( g(x) = (x + 2)^3 \) for each chosen value of \( x \). For each x-value:- \( x = -4 \): \( g(-4) = (-4 + 2)^3 = (-2)^3 = -8 \)- \( x = -3 \): \( g(-3) = (-3 + 2)^3 = (-1)^3 = -1 \)- \( x = -2 \): \( g(-2) = (-2 + 2)^3 = 0^3 = 0 \)- \( x = -1 \): \( g(-1) = (-1 + 2)^3 = 1^3 = 1 \)- \( x = 0 \): \( g(0) = (0 + 2)^3 = 2^3 = 8 \)- \( x = 1 \): \( g(1) = (1 + 2)^3 = 3^3 = 27 \)- \( x = 2 \): \( g(2) = (2 + 2)^3 = 4^3 = 64 \)

Step 3: Create the Table

Now, let's create a table of the x-values and their corresponding y-values from the calculations above: | x | g(x) | |----|------| | -4 | -8 | | -3 | -1 | | -2 | 0 | | -1 | 1 | | 0 | 8 | | 1 | 27 | | 2 | 64 |

Step 4: Sketch the Graph

Using the table of values, plot the points \((-4, -8), (-3, -1), (-2, 0), (-1, 1), (0, 8), (1, 27), (2, 64)\) on a graph. Draw a smooth curve through these points. Since the function is a cubic function of the form \( (x+2)^3 \), the graph will have a typical S-shape of a cubic graph, starting low, passing through zero, and then rising steeply.

Key Concepts

Graphing Functions

Graphing functions can seem daunting at first, but it's all about plotting points on a graph to visualize how a function behaves. For a cubic function like \(g(x) = (x+2)^3\), the graph usually shows an S-shape. This signature shape is important because it helps us understand the growth rate of the function.When graphing, follow these general steps:

• Identify key points where the function changes direction. These include any zeros (where the function crosses the x-axis), local maxima, and minima.
• Plot calculated points on the graph, and connect them smoothly.

In our specific case of \((x+2)^3\), notice how as x increases past zero, the graph steeply rises. This reflects the rapid growth nature of cubic functions. The graph starts from negative values, crosses through zero, and then climbs upwards swiftly, showing what we call as inflection point behavior.

Table of Values

Creating a table of values is an essential step in plotting functions accurately, as it helps translate the function's equation into concrete data. When forming this table, you are simply computing the corresponding y-values for various x-values, giving you points to plot on your graph.Here's what to remember:

• Choose a range of x-values that is representative of your function's behavior. Often, this range is centered around points of interest, such as zeros or inflection points.
• Compute y-values using your function's formula for each chosen x-value. In our exercise, these include: \[ \begin{aligned} & x = -4, \quad g(x) = -8, \ & x = -3, \quad g(x) = -1, \ & x = -2, \quad g(x) = 0, \ & x = -1, \quad g(x) = 1, \ & x = 0, \quad g(x) = 8, \ & x = 1, \quad g(x) = 27, \ & x = 2, \quad g(x) = 64 \end{aligned} \]

As you see, the table of values provides the necessary foundation to create a visual representation of the function.

Function Evaluation

Function evaluation is like asking a function a question and waiting for it to "respond" with an answer. In simpler terms, it involves taking a particular input (an x-value) and finding out what the output (the corresponding y-value) will be. For our function \(g(x) = (x+2)^3\), the process involves a few straightforward steps:

• Start with the given x-value from your set range, such as \(x = -4\).
• Substitute this value into the function equation \(g(x) = (x + 2)^3\).
• Simplify the equation to arrive at the resulting y-value, \(g(x)\).

For example, using \(x = 0\), substitute to find \(g(0) = (0 + 2)^3 = 8\). This step-by-step approach aids in methodically plotting each point of the cubic function as seen in our graph. Understanding each small evaluation helps to see the bigger picture of how functions grow.