Problem 27 - Calculus Volume 1

Question

For the following exercises, set up a table to sketch the graph of each function using the following values: \(x=-3,-2,-1,0,1,2,3\) $$ f(x)=x^{3} $$

Step-by-Step Solution

Answer

Calculate the cube of each given x and plot it on a graph to sketch the curve.

Step 1: Create the Table Format

First, set up a table with two columns. The first column is for the values of \( x \), and the second column is for the corresponding values of \( f(x) = x^3 \). Prepare the table with rows for each \( x \) in \( \{-3, -2, -1, 0, 1, 2, 3\} \).

Step 2: Calculate f(x) for Each x Value

For each \( x \) value, calculate \( f(x) = x^3 \). Fill in the second column of the table with these results:- For \( x = -3 \), \( f(x) = (-3)^3 = -27 \)- For \( x = -2 \), \( f(x) = (-2)^3 = -8 \)- For \( x = -1 \), \( f(x) = (-1)^3 = -1 \)- For \( x = 0 \), \( f(x) = 0^3 = 0 \)- For \( x = 1 \), \( f(x) = 1^3 = 1 \)- For \( x = 2 \), \( f(x) = 2^3 = 8 \)- For \( x = 3 \), \( f(x) = 3^3 = 27 \).

Step 3: Complete the Table

Insert all calculated \( f(x) \) values into the table:| \( x \) | \( f(x) \) ||---|---|| -3 | -27 || -2 | -8 || -1 | -1 || 0 | 0 || 1 | 1 || 2 | 8 || 3 | 27 |

Step 4: Sketch the Graph

Using the table, plot each \( (x, f(x)) \) point on a Cartesian plane. Connect these points with a smooth curve, as the function \( f(x) = x^3 \) creates a continuous curve. The graph will be symmetric around the origin, showing an increasing trend in the cubic shape.

Key Concepts

Table of Values

Creating a table of values is a fundamental step in graphing functions. It helps you visualize how the function behaves at specific points. To start this process, you set up a table with two columns. The first column holds the x values you are interested in. In our example, these are \(-3, -2, -1, 0, 1, 2, 3\). The second column is for the corresponding output values of the function, which in our case is \(f(x) = x^3\). This approach provides a clear visualization of how the cubic function \(f(x)\) evolves as \(x\) varies. Tables like this are a powerful tool to comprehend the relationship between the input and output, simplifying the path to graphing the function.

Function Values Calculation

Once your table format is ready, calculating the function's values for each chosen \(x\) is the next focus. Here, you evaluate the function \(f(x) = x^3\) for each \(x\) value. This involves raising each \(x\) value to the power of three.

• For \(x = -3\), \(f(x) = (-3)^3 = -27\)
• For \(x = -2\), \(f(x) = (-2)^3 = -8\)
• For \(x = -1\), \(f(x) = (-1)^3 = -1\)
• For \(x = 0\), \(f(x) = 0^3 = 0\)
• For \(x = 1\), \(f(x) = 1^3 = 1\)
• For \(x = 2\), \(f(x) = 2^3 = 8\)
• For \(x = 3\), \(f(x) = 3^3 = 27\)

This list shows how the function's output responds to changes in input value. Carefully calculating values ensures accuracy in the graphing stage.

Graph Sketching

With the table of values complete, it's time to bring the function to life on a graph. Start by plotting each point from your table on a Cartesian plane. This involves marking points for each \( (x, f(x)) \) pair, such as \((-3, -27)\) and \( (3, 27)\) on the graph.To connect the points, draw a smooth, continuous curve through them, reflecting the natural shape of the cubic function \( f(x) = x^3\). This curve will appear symmetric about the origin, showing an increasing trend moving from left to right across the axis as characterized by cubic functions.Adding a graph gives you a visual understanding of how the function behaves. You can see the steepness increase as x moves away from zero, and the symmetry emphasizes its graphical properties. This approach offers a concrete insight into mathematical concepts through visual representation.