Problem 22 - College Algebra with Modeling and Visualization

Question

Exercises \(19-32:\) Graph the linear function by hand. Identify the slope and y-intercept. $$ f(x)=3-x $$

Step-by-Step Solution

Answer

The slope is \(-1\) and the y-intercept is \((0, 3)\).

Step 1: Rewrite the Function

The function given is \( f(x) = 3 - x \), which can be rewritten in the slope-intercept form \( y = mx + b \) as \( y = -x + 3 \). This makes it easier to identify the slope and y-intercept.

Step 2: Identify the Slope and Y-intercept

In the rewritten equation \( y = -x + 3 \), the slope \( m \) is \(-1\) and the y-intercept \( b \) is \(3\). This means the line decreases by \(1\) unit vertically for every \(1\) unit it moves horizontally to the right.

Step 3: Plot the Y-intercept

Start by plotting the y-intercept on the coordinate grid. The y-intercept is the point \((0, 3)\). Place a point on the y-axis at \(3\).

Step 4: Use the Slope to Find Another Point

From the y-intercept \((0, 3)\), use the slope \(-1\) to find another point. Move one unit to the right (positive x-direction) and one unit down (negative y-direction) to get the point \((1, 2)\).

Step 5: Draw the Line

Draw a straight line through the points \((0, 3)\) and \((1, 2)\). This line extends infinitely in both directions and represents the graph of the function \( f(x) = 3 - x \).

Step 6: Verify with Another Point

To ensure accuracy, choose another x-value, such as \( x = 2 \). Substitute it into the function: \( f(2) = 3 - 2 = 1 \). The point \((2, 1)\) should be on the line, and plotting it confirms the line's correctness.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a simple way to express the equation of a line. It is written as \( y = mx + b \).

• \( y \) represents the dependent variable, usually represented by the y-coordinate on a graph.
• \( m \) is the slope of the line, which shows how steep the line is.
• \( x \) is the independent variable, shown by the x-coordinate on a graph.
• \( b \) is the y-intercept, the value of \( y \) when \( x \) is 0.

To convert a linear function like \( f(x) = 3 - x \) into slope-intercept form, rearrange it to look like \( y = mx + b \). This way, it becomes easier to spot both the slope and the y-intercept right away. For instance, \( f(x) = 3 - x \) when rewritten in slope-intercept form is \( y = -x + 3 \). By examining this form, we can see the slope and intercept clearly.

Slope of a Line

The slope of a line is a measure that tells you how much the line tilts, either upwards or downwards, as you move along the x-axis. It is denoted by \( m \) in the slope-intercept form \( y = mx + b \).

• A positive slope means the line tilts upward from left to right.
• A negative slope means the line slopes downward from left to right.
• If the slope is zero, the line is perfectly horizontal.
• An undefined slope indicates a vertical line.

For the function \( f(x) = 3 - x \), which becomes \( y = -x + 3 \) in slope-intercept form, the slope \( m \) is \(-1\). This indicates that for every step you take to the right on the x-axis, you move one step down on the y-axis. The line's downward tilt shows the negative slope, leading to a decrease in y-values as x-values increase.

Y-intercept

The y-intercept is a crucial point where a line crosses the y-axis. It represents the value of \( y \) when \( x \) is zero. In the slope-intercept form \( y = mx + b \), \( b \) is the y-intercept.

• The y-intercept gives you a starting point to help draw the graph of a line.
• The point is located at \((0, b)\) on the coordinate grid.

For our example, \( f(x) = 3 - x \), which can be rewritten as \( y = -x + 3 \), the y-intercept \( b \) is \( 3 \). This means when \( x = 0 \), \( y = 3 \), so the line crosses the y-axis at the point \((0, 3)\). Start plotting your graph by putting a point on the y-axis at 3. This y-intercept provides a reference point to draw the rest of the line using the slope.