Problem 9 - College Algebra with Modeling and Visualization

Question

Graph \(y=f(x)\) by hand by first plotting points to determine the shape of the graph. $$ f(x)=4-x $$

Step-by-Step Solution

Answer

The graph is a straight line with a negative slope, passing through points like (0, 4), (1, 3), and (2, 2).

Step 1: Understanding the Function

The function given is a linear equation \(f(x) = 4 - x\). This can be rewritten as \(y = 4 - x\) and is in the form \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept.

Step 2: Identifying the Slope and Intercept

Here, the slope \(m\) is \(-1\) and the y-intercept \(c\) is \(4\). This means the line will decrease as \(x\) increases and will cross the y-axis at \(4\).

Step 3: Choose Points to Plot

Select some values for \(x\) to find corresponding points on the line. Choose at least three points to ensure accuracy in drawing the line (e.g., \(x = 0, 1, 2\)).

Step 4: Calculate Corresponding \(y\)-Values

For each chosen \(x\)-value, substitute it into the equation to find the corresponding \(y\)-value:- If \(x = 0\), then \(y = 4 - 0 = 4\).- If \(x = 1\), then \(y = 4 - 1 = 3\).- If \(x = 2\), then \(y = 4 - 2 = 2\).

Step 5: Plot Points on the Graph

Plot the points \((0, 4)\), \((1, 3)\), and \((2, 2)\) on a Cartesian plane. Ensure the scale is consistent and each point is accurately plotted.

Step 6: Draw the Line

Using a ruler, connect the points with a straight line extending across the plane. This line represents the graph of \(f(x) = 4 - x\).

Key Concepts

Plotting Points

Plotting points is a critical step when graphing linear functions. It helps to visualize the line that the linear equation represents. To begin, we need to select some values for the variable, typically "\(x\)", to calculate the respective \(y\)-values for each. These pairs \((x, y)\) will be our points to plot on the graph.

Caring for accuracy, it's advisable to choose at least three distinct points. For the function \(f(x) = 4 - x\), we can choose some simple \(x\)-values like \(0, 1,\) and \(2\).

• When \(x = 0\), substituting it into the function gives us \(y = 4 - 0 = 4\), forming the point \((0, 4)\).


• Similarly, for \(x = 1\), \(y = 4 - 1 = 3\) gives the point \((1, 3)\).


• And for \(x = 2\), \(y = 4 - 2 = 2\), resulting in the point \((2, 2)\).

Once you've calculated these points, plot them on a graph. Ensure every plot is precise by using a consistent scale.

Slope and Y-Intercept

In linear equations, the slope and y-intercept are two fundamental components that determine the position and angle of the line on a graph. A linear equation is usually represented in the form \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept.

For the equation \(f(x) = 4 - x\), which can be rewritten as \(y = -x + 4\), you'll notice:

• The slope \(m = -1\). This negative slope tells us that the line will fall or decrease as you move from left to right across the graph.


• The y-intercept \(c = 4\) is where the line crosses the y-axis. This happens at the point \((0, 4)\).

Understanding the slope allows you to ascertain how sharp or gradual the line slopes, while the y-intercept provides an exact starting point on the y-axis.

Linear Equations

Linear equations are expressions where the variables are raised only to the power of one. They describe straight lines on a graph. An essential characteristic worth noting is their format, \(y = mx + c\), which outlines a clear linear relationship between \(x\) and \(y\).

For our function, \(f(x) = 4 - x\), rewriting that gives us \(y = -x + 4\). This equation confirms that when plotted, it will form a straight line. Let's see the reasons why linear equations appear as lines on a graph:

• Because the slope \(m\) is constant, indicating a consistent rise over run, or gradient, along the entire length of the line.


• The y-intercept \(c\) provides a precise point where the line meets the y-axis, ensuring the graph's origin.

Linear equations are crucial within various fields, from mathematics to economics, as they can model data and relationships reliably with their straightforward, predictable nature.