Problem 23 - Calculus Volume 1

Question

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

Find the values of \( f(x) \) and plot the points to sketch the graph.

Step 1: Identify the Function

The given function is a linear function given by \( f(x) = 3x - 6 \). It is in the form of \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Step 2: Create a Table of Values

We will create a table of values for the given \( x \)-values: \(-3, -2, -1, 0, 1, 2, 3\). Calculate \( f(x) \) for each of these values and fill them in the table.| \( x \) | \( f(x) = 3x - 6 \) ||--------|------------------|| -3 | || -2 | || -1 | || 0 | || 1 | || 2 | || 3 | |

Step 3: Calculate \( f(x) \) for Each \( x \)-Value

Using the function \( f(x) = 3x - 6 \), calculate the value of \( f(x) \) for each \( x \):- For \( x = -3 \), \( f(-3) = 3(-3) - 6 = -9 - 6 = -15 \).- For \( x = -2 \), \( f(-2) = 3(-2) - 6 = -6 - 6 = -12 \).- For \( x = -1 \), \( f(-1) = 3(-1) - 6 = -3 - 6 = -9 \).- For \( x = 0 \), \( f(0) = 3(0) - 6 = 0 - 6 = -6 \).- For \( x = 1 \), \( f(1) = 3(1) - 6 = 3 - 6 = -3 \).- For \( x = 2 \), \( f(2) = 3(2) - 6 = 6 - 6 = 0 \).- For \( x = 3 \), \( f(3) = 3(3) - 6 = 9 - 6 = 3 \).

Step 4: Fill in the Table of Values

Now, fill in the values calculated from Step 3 into the table:| \( x \) | \( f(x) = 3x - 6 \) ||--------|------------------|| -3 | -15 || -2 | -12 || -1 | -9 || 0 | -6 || 1 | -3 || 2 | 0 || 3 | 3 |

Step 5: Sketch the Graph

Using the completed table of values, plot the points \((-3, -15), (-2, -12), (-1, -9), (0, -6), (1, -3), (2, 0), (3, 3)\) on a coordinate plane. Connect the points with a straight line, showcasing the linear function.

Key Concepts

Graph Sketching

Graph sketching is a valuable skill that can help visualize mathematical relationships, such as linear functions. Start by plotting points you have calculated using your table of values. These points are crucial, as they represent input-output pairs for your function. For the function \( f(x) = 3x - 6 \), you would begin by plotting points like \((-3, -15)\) and \((3, 3)\).Once your points are in place, draw a straight line through them. Linear functions, by definition, connect in a perfectly straight line when plotted in a coordinate plane. This visual representation not only helps in understanding but also in interpreting the function's behavior over different values of \( x \). This way, you can predict function values without always calculating them directly.

Table of Values

A table of values is an essential tool in solving and graphing linear equations. It helps you organize values and their corresponding outputs efficiently, making calculations straightforward. For the equation \( f(x) = 3x - 6 \), the table provides a clear picture of how \( f(x) \) changes as \( x \) changes.

• Identify each \( x \) value you will use, e.g., \(-3, -2, -1, 0, 1, 2, 3\).
• Calculate the corresponding \( f(x) \) for each \( x \) using the function.
• Write down these pairs, aiding in plotting the graph accurately.

Using a table of values, you can systematically trace how the input \( x \) affects the output \( f(x) \), thus better understanding the linear relationship at hand.

Coordinate Plane

The coordinate plane is the stage on which graphing unfolds, providing a visual grid to plot linear equations. It consists of two axes: the horizontal \( x \)-axis and the vertical \( y \)-axis, intersecting at the origin \((0,0)\). Each point on the plane corresponds to an \((x, y)\) pair.When graphing the function \( f(x) = 3x - 6 \), begin by carefully marking each point from your table of values onto this plane. The consistent spacing of axes ensures that each line segment accurately reflects changes dictated by the function's formula. The coordinate plane thus becomes a powerful visual aid, helping you to see the relationship between \( x \) and \( y \) clearly and precisely.

Slope and Intercept

Understanding slope and intercept is crucial to mastering linear functions. The general form for these functions is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Slope (\( m \))

The slope indicates the steepness of the line and the direction it moves across the graph. For \( f(x) = 3x - 6 \), the slope is \( 3 \), meaning for every unit increase in \( x \), \( y \) increases by \( 3 \).

Y-Intercept (\( b \))

The y-intercept is where the function crosses the y-axis, occurring when \( x = 0 \). In our function, the y-intercept is \(-6\), representing the point \((0, -6)\). This is where the line meets the y-axis, providing crucial information about the function's starting value on the graph.These two parameters define the line's trajectory and position within the coordinate plane, offering valuable insight into the function's dynamics.