Problem 5 - 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

Create a table of values from the function, plot the points on a graph, and connect them to form a line.

Step 1: Understanding the Function

The function given is a linear function: \( f(x) = -x + 3 \). This type of function results in a straight line when graphed. The function has a slope of \(-1\) and a y-intercept of \(3\).

Step 2: Creating a Table of Values

To create a table of values, select values within the specified range \( -3 \leq x \leq 3 \). It's helpful to pick integer values to simplify calculations. This includes \(x = -3, -2, -1, 0, 1, 2, 3\).

Step 3: Calculating Values for the Table

Plug each selected \(x\) value into the function \( f(x) = -x + 3 \) to find the corresponding \(f(x)\) or \(y\) values:- When \(x = -3\), \(f(-3) = -(-3) + 3 = 6\).- When \(x = -2\), \(f(-2) = -(-2) + 3 = 5\).- When \(x = -1\), \(f(-1) = -(-1) + 3 = 4\).- When \(x = 0\), \(f(0) = -(0) + 3 = 3\).- When \(x = 1\), \(f(1) = -(1) + 3 = 2\).- When \(x = 2\), \(f(2) = -(2) + 3 = 1\).- When \(x = 3\), \(f(3) = -(3) + 3 = 0\).

Step 4: Compiling the Table of Values

The calculated values are:\(\begin{array}{c|c} x & f(x) \\hline -3 & 6 \ -2 & 5 \ -1 & 4 \ 0 & 3 \ 1 & 2 \ 2 & 1 \ 3 & 0 \\end{array}\)

Step 5: Sketching the Graph

Plot the points from the table of values on a graph. For each \((x, f(x))\) pair, place a point at the corresponding coordinates. Connect these points using a straight line since this is a linear function. Label the axes and ensure the x-values range from \(-3\) to \(3\) and y-values from \(0\) to \(6\).

Step 6: Finalizing the Graph

Review the plotted points to ensure they follow the intended linear path described by the function \(f(x) = -x + 3\), with a downward slope, intersecting the y-axis at \(3\). Check that each point from the table is accurately represented on the graph.

Key Concepts

Understanding Linear Equations

Linear equations, like the one from the exercise, take the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. These equations are called "linear" because they graph as straight lines. The slope \(m\) indicates the tilt or steepness of the line, determining how the line moves in the coordinate plane. If \(m\) is positive, the line rises as you move from left to right across the graph. Alternatively, a negative \(m\) means the line falls. The y-intercept \(b\), on the other hand, marks where the line crosses the y-axis. In our particular function, \(f(x) = -x + 3\), the slope is \(-1\), indicating a downward slant, while the y-intercept is \(3\), meaning the line passes through the y-axis at 3.

Making a Table of Values

Creating a table of values is a simple step to help plot linear equations accurately. It involves substituting selected values for \(x\) into the equation to find corresponding \(y\) values. Ideally, choose values within the range specified by the problem to simplify drawing the graph. In our exercise, the range is \(-3 \leq x \leq 3\).

Steps to create a table of values:

• Pick integer values for \(x\) within the given range; they simplify calculations.
• Substitute each chosen \(x\) into the equation \(f(x) = -x + 3\).
• Solve for \(f(x)\) to obtain corresponding \(y\) values.
• List the \((x, y)\) pairs in a table format to prepare for plotting.

This table helps to establish a clear set of coordinates to graph the equation efficiently.

Explaining Slope and Y-Intercept

Slope and y-intercept are foundational concepts when graphing linear equations. The slope \(m\) determines the direction and angle of your line. It tells you how much \(y\) changes as \(x\) increases by 1.
Key Points about Slope:

• A slope of \(-1\) means for every unit increase in \(x\), \(y\) decreases by 1 unit.
• Slope is visually represented as the rise over run in the line.

The y-intercept \(b\) is the starting point of your line on the y-axis.
Understanding Y-Intercept:

• \(b = 3\) means when \(x = 0\), \(y = 3\).
• This is the point where your line crosses the y-axis, which is visible when graphing your equation.

The slope and y-intercept together define the precise path of the line on a graph.

Plotting Points on a Graph

Plotting points on a graph is the final step in visualizing a linear equation. Using the table of values, you can place points with clear coordinates. Here are some steps for effective plotting:

• First, draw your axes and label them, ensuring they cover the range you need, like \(-3\) to \(3\) for \(x\) and up to \(6\) for \(y\).
• Next, use the table of values to locate each \((x, y)\) point on the graph.
• When all points are plotted, use a ruler to connect them with a straight line, confirming the linear nature of the equation.
• Finally, review the graph to ensure it aligns with the function's slope and y-intercept.

By following these steps, you transform abstract numbers into a coherent visual story, providing a clear understanding of the linear equation's behavior.