Problem 23 - Calculus Volume 1

Question

For the following exercises, set up a table to sketch the graph of each function using the following values: \(x=-3,-2,-1,0,1,2,3\) $$ f(x)=3 x-6 $$

Step-by-Step Solution

Answer

Use a table to calculate values of f(x) at given x, then plot these to sketch the graph.

Step 1: Understand the Function

The given function is a linear equation in the form of \( f(x) = 3x - 6 \). This means for every \( x \), the y-value or \( f(x) \) is equal to \( 3 \times x - 6 \).

Step 2: Create a Table

Set up a table with two columns: one for \( x \) values and one for \( f(x) \) values. You'll calculate \( f(x) \) for each \( x \) value provided.

Step 3: Calculate f(x) for x = -3

Substitute \( x = -3 \) into the equation, \( f(x) = 3(-3) - 6 = -9 - 6 = -15 \). Enter \(-3\) for x and \(-15\) for f(x) in the table.

Step 4: Calculate f(x) for x = -2

Substitute \( x = -2 \) into the equation, \( f(x) = 3(-2) - 6 = -6 - 6 = -12 \). Enter \(-2\) for x and \(-12\) for f(x) in the table.

Step 5: Calculate f(x) for x = -1

Substitute \( x = -1 \) into the equation, \( f(x) = 3(-1) - 6 = -3 - 6 = -9 \). Enter \(-1\) for x and \(-9\) for f(x) in the table.

Step 6: Calculate f(x) for x = 0

Substitute \( x = 0 \) into the equation, \( f(x) = 3(0) - 6 = 0 - 6 = -6 \). Enter \(0\) for x and \(-6\) for f(x) in the table.

Step 7: Calculate f(x) for x = 1

Substitute \( x = 1 \) into the equation, \( f(x) = 3(1) - 6 = 3 - 6 = -3 \). Enter \(1\) for x and \(-3\) for f(x) in the table.

Step 8: Calculate f(x) for x = 2

Substitute \( x = 2 \) into the equation, \( f(x) = 3(2) - 6 = 6 - 6 = 0 \). Enter \(2\) for x and \(0\) for f(x) in the table.

Step 9: Calculate f(x) for x = 3

Substitute \( x = 3 \) into the equation, \( f(x) = 3(3) - 6 = 9 - 6 = 3 \). Enter \(3\) for x and \(3\) for f(x) in the table.

Step 10: Sketch the Table

Using the values calculated, you can create a table like this: | x | f(x) | |:-:|:---:| | -3 | -15 | | -2 | -12 | | -1 | -9 | | 0 | -6 | | 1 | -3 | | 2 | 0 | | 3 | 3 |

Step 11: Conclusion

With the table complete, you can now use these points to sketch the graph of the linear function.

Key Concepts

Function Table

A function table is a valuable tool when working with linear equations, allowing you to organize and calculate values systematically. It's a table with two columns: one for the input values (often denoted as \(x\)) and another for the output values (the results of the function, \(f(x)\)).

To fill in a function table, you follow a series of easy steps:

• Identify the linear equation you are dealing with. Here, it's \(f(x) = 3x - 6\).
• List the \(x\) values you need to calculate, which in this exercise are \(-3, -2, -1, 0, 1, 2, 3\).
• Substitute each \(x\) value into the function to compute \(f(x)\). This means solving \(3x - 6\) for each \(x\) value.
  

For example, for \(x = -3\), the output \(f(x)\) is \(-15\). You continue these calculations for each \(x\) value and enter them into the table.

Using a function table not only helps in finding the necessary points for graph sketching but also provides a visual method to see how the \(x\) values affect the function's output.

Graph Sketching

Graph sketching involves drawing the pictorial representation of an equation, particularly how changes in the input (\(x\)) affect the output (\(f(x)\) or \(y\)). In the context of a linear equation, this graph will always be a straight line.

To sketch a graph from a linear equation, follow these steps:

• Utilize the function table to obtain some coordinate points (pairs of \(x\) and \(f(x)\) values) such as \((-3, -15)\), \((-2, -12)\), and so on.
• Plot these points on a coordinate plane, which consists of an \(x\)-axis (horizontal) and a \(y\)-axis (vertical).
• Once your points are plotted, use a ruler to draw a line through these points. The line should extend in both directions as linear functions continue infinitely.
  

Graph sketching with a clear function table helps to visually confirm the function's behavior and check the accuracy of your calculations without strictly solving everything algebraically. This practical approach is particularly useful in understanding linear relationships and graph transformations.

Coordinate Points

In mathematics, coordinate points are used to identify the position of any point on a plane. Each point is defined by an ordered pair \((x, y)\) where \(x\) is the horizontal value and \(y\) is the vertical value.

For graphing linear functions, obtaining coordinate points from a function table is essential. For each \(x\) value you input, you calculate \(y\) as \(f(x)\), forming your coordinate points.

Here's how you generate and use coordinate points:

• Each solution of the function \(f(x) = 3x - 6\) gives you a new coordinate point.
• When \(x\) is \(-3\), \(f(x)\) is \(-15\), creating the point \((-3, -15)\).
• Similarly, when \(x\) is \(3\), \(f(x)\) results in \(3\), forming the coordinate point \((3, 3)\).

Once all points are calculated, graph them on the plane. Properly plotting these predetermined points and connecting them is how the graph of a linear function is formed, showcasing the direct correlation between \(x\) and \(f(x)\).

Understanding coordinate points and how they are used simplifies graphing and aids in visualizing mathematical relationships effectively.