Problem 9 - Algebra and Trigonometry

Question

Sketch the graph of the function by first making a table of values. \(f(x)=-x+3, \quad-3 \leq x \leq 3\)

Step-by-Step Solution

Answer

Plot points from the table and connect them to form a line.

Step 1: Create a List of Values for x

Create a table with values of \(x\) ranging from \(-3\) to \(3\). This will include the numbers: \(-3, -2, -1, 0, 1, 2,\) and \(3\).

Step 2: Calculate Corresponding f(x) Values

Use the function formula \(f(x) = -x + 3\) to calculate \(f(x)\) for each value of \(x\) in the list created in Step 1. For example, if \(x = -3\), then \(f(-3) = -(-3) + 3 = 6\). Repeat this calculation for each \(x\) value.

Step 3: Complete the Table of Values

Fill in the table with the calculated \(f(x)\) values from Step 2:\[\begin{array}{c|c}x & f(x) \\hline-3 & 6 \-2 & 5 \-1 & 4 \0 & 3 \1 & 2 \2 & 1 \3 & 0 \\end{array}\]

Step 4: Plot the Points on a Graph

Using the table of values, plot the points \((-3, 6), (-2, 5), (-1, 4), (0, 3), (1, 2), (2, 1), (3, 0)\) on the Cartesian plane.

Step 5: Draw the Graph

Connect the plotted points with a straight line, as the function \(f(x) = -x + 3\) is a linear function. This line represents the graph of the function within the interval \(-3 \leq x \leq 3\).

Key Concepts

Table of Values

To graph a linear function like \( f(x) = -x + 3 \), creating a table of values is essential. It acts as a map that guides you in plotting the graph accurately.
Start by selecting several values for \( x \). In our example, we choose \( x \) values ranging from \(-3\) to \(3\). These are

• \(-3\)
• \(-2\)
• \(-1\)
• 0
• 1
• 2
• 3

Next, calculate \( f(x) \) using the formula \( f(x) = -x + 3 \) for each selected \( x \).
This will give you a corresponding \( f(x) \) value for each \( x \), such as \( f(-3) = 6 \), and \( f(3) = 0 \).
Finally, compile these into a table, creating pairs \((x, f(x))\). This table will be your essential tool for the next steps of plotting the graph.

Cartesian Plane

The Cartesian plane is the stage where the magic of graphing unfolds. It's a flat surface defined by two perpendicular axes: the horizontal \( x \)-axis and the vertical \( y \)-axis.
Think of it as a giant grid, where each point is specified by a pair of coordinates \((x, y)\). The point \((0, 0)\) is called the origin, and it sits where the two axes meet.
When you're ready to graph your function, you will plot each paired value from your table onto this grid.

• The first number in the pair is the \( x \)-coordinate, which tells you how far along the \( x \)-axis your point is.
• The second number is the \( y \)-coordinate, which dictates the point's position along the \( y \)-axis.

Learning to interpret and navigate the Cartesian plane efficiently is key to mastering graphing.

Linear Equations

Linear equations are mathematical expressions that form a straight line when graphed. The standard form of a linear equation is often written as \( y = mx + b \).
In our exercise, the function \( f(x) = -x + 3 \) is a linear equation, where

• \( m \) (the slope) is \(-1\), indicating the line falls as \( x \) increases
• \( b \) (the y-intercept) is \(3\), showing where the line crosses the \( y \)-axis

Linear equations are easy to graph because any two points will allow you to draw the line.
Calculating a few more points helps ensure accuracy, especially over larger graphs. This makes linear equations a vital tool for not only mathematics but also practical applications in fields like engineering and economics.

Plotting Points

Plotting points is like playing connect-the-dots, but with a mathematical twist. Using the table of values, each \((x, f(x))\) pair needs to be represented as a dot on the Cartesian plane.
Start by finding the \( x \) value on the horizontal axis, and then move vertically to the \( y \) value.
Plot each point carefully, such as \((-3, 6)\) or \((1, 2)\).

• Make sure the points are in the correct position according to your table.
• Use a ruler or a straight tool to connect dots once all points are plotted.

The goal is to draw a straight line that extends beyond the last points plotted if needed.
This results in an accurate graphical representation of your linear function, showing the relationship between \( x \) and \( f(x) \) clearly.