Problem 44 - College Algebra

Question

Give the slope and y-intercept of each line whose equation is given. Then graph the line. $$y=\frac{3}{4} x-3$$

Step-by-Step Solution

Answer

The slope of the line is \(\frac{3}{4}\) and the y-intercept is -3.

Step 1: Identify the Slope

In the slope-intercept form equation \(y = mx + c\), 'm' gives the slope. Here, \(m = \frac{3}{4}\). So, the slope of the line is \(\frac{3}{4}\).

Step 2: Identify the Y-Intercept

In the slope-intercept form equation \(y = mx + c\), 'c' gives the y-intercept. Here, \(c = -3\). So, the y-intercept of the line is -3.

Step 3: Graph the Line

Start by plotting the y-intercept (-3) on the y-axis. From this point, use the slope (\(\frac{3}{4}\)) to identify another point on the line. The slope is the change in 'y' for a one unit increase in 'x', so from the y-intercept, move up 3 units and to the right 4 units and mark this point. Draw a straight line through these two points for the graph of the line.

Key Concepts

Graphing Lines

When graphing lines, we use the equation of the line to provide a visual representation on a coordinate plane.

A line can be graphed using its slope and y-intercept, which can be found in the slope-intercept form of an equation, typically written as \(y = mx + c\). This form clearly shows the slope \(m\) and the y-intercept \(c\).

To graph:

• Start by plotting the y-intercept on the y-axis. This point, \(c\), is where the line crosses the y-axis, which means \(x = 0\).
• From the y-intercept, use the slope \(m\) to determine another point on the line. Remember, the slope represents the rise over run.
• Connect these points with a straight line, extending it across the graph to see its full trajectory.

This visual process helps understand how different linear equations translate into distinctive slopes and positions on the graph.

Slope

The slope of a line in the slope-intercept form \(y = mx + c\) is denoted by \(m\).

The slope shows how steep the line is and in which direction it tilts.
Here, the slope is \(\frac{3}{4}\), which means:

• The line ascends, indicating a positive slope.
• For every 4 units you move horizontally (right) on the x-axis, you move 3 units vertically (up) on the y-axis.

Understanding slope is crucial:

• It shows the rate of change between the variables.
• A positive slope suggests an upward trend, while a negative slope would imply a downward trend.
• The greater the slope value, the steeper the line.

Knowing how to interpret slope helps in predicting how changes in one variable affect the other.

Y-Intercept

The y-intercept is the point where the graph of the line crosses the y-axis. In the equation \(y = mx + c\), the y-intercept is given by \(c\). For our equation, \(c = -3\), which translates to the point \((0, -3)\).

To understand why the y-intercept is important:

• It's the value of \(y\) when \(x = 0\).
• This point acts as a starting point on the graph for drawing the rest of the line based on the slope.

In practical terms, the y-intercept gives us a foundational point for any linear equation, helping easily plot the line on a graph.
Knowing the y-intercept aids in:

• Quickly visualizing where the line will cross the y-axis.
• Allowing for fast graph setup without needing to solve for points manually.

This feature simplifies graphing tasks and helps in understanding the behavior of linear equations.