Problem 68 - Introductory and Intermediate Algebra for College Students 4th

Question

Describe how to graph a line using the slope and \(y\) -intercept. Provide an original example with your description.

Step-by-Step Solution

Answer

To graph a line with slope and y-intercept, first understand what each mean: the slope is the steepness of the line (vertical change divided by horizontal change), and the y-intercept is where the line crosses the y-axis. Using the equation of a line, \(y = mx + b\), draw a point at the y-intercept, then use the slope to plot the remaining points and draw the line. For example, with m=3, b=2, start at the point (0,2) and from there, every following point can be found by going up 3 units and to the right 1 unit.

Step 1: Definition of Slope and y-intercept

The slope is a measure that indicates the inclination, or steepness, of a line, and it is the vertical change (or rise) over the horizontal change (or run). The y-intercept is the point where the line crosses the y-axis. For the equation of a line, \(y = mx + b\), \(m\) is the slope, and \(b\) is the y-intercept.

Step 2: Identify the values of m and b

Decide on values for the slope (m) and y-intercept (b) to be used in the equation of the line. For example, m=3 and b=2.

Step 3: Plot y-intercept

The y-intercept is the point where the line crosses the y-axis, so we find the value of 2 on the y-axis and make a point.

Step 4: Determine and plot next points using the slope

From the y-intercept, we use the slope to find the next points on the line. Since the slope is 3 (or 3/1), from the y-intercept, each point on the line can be found by going up 3 units (the rise) and to the right 1 unit (the run). We can do this several times to plot several points.

Step 5: Draw the Line

Finally, after the points have been plotted, a line that passes through all these points is drawn. This is our desired graph.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is key to graphing a line in algebra. This form is expressed as the equation \(y = mx + b\), where \(m\) represents the slope, and \(b\) denotes the y-intercept.

The slope, \(m\), indicates how slanted the line is and is calculated by the rise over the run—that is, how much the line goes up (or down) for each unit it goes right (or left). Positive slopes rise to the right, and negative slopes fall to the right. A zero slope means the line is horizontal, and an undefined slope indicates a vertical line.

The y-intercept, \(b\), is simply the point where the line crosses the y-axis. This occurs when \(x = 0\), giving the graph a starting point without any calculations necessary.

For example, if we have an equation of a line like \(y = 2x + 3\), the slope \(m\) is 2, which means for every one unit you go right along the x-axis, you move up 2 units along the y-axis. The y-intercept \(b\) is 3, which means the line crosses the y-axis at the point \((0, 3)\).

Plotting the line starts by marking the y-intercept on the graph and then using the slope to find additional points.

Y-Intercept

Understanding the Starting Point

The y-intercept is an integral component of the line equation in the slope-intercept form. It's where your line will make its entrance on the graph. Symbolized by \(b\) in the equation \(y = mx + b\), it indicates the exact spot on the y-axis where the line will pass through.

This point on the graph can be found where \(x\) is zero. For instance, if your equation is \(y = -4x + 1\), the y-intercept is \((0, 1)\). That's your line's starting point on the graph.

When plotting a line on a graph, always start by marking the y-intercept. It provides a reference point from which you can use the slope to determine the direction and steepness of the line as it stretches across the graph.

Plotting Points

Bringing the Line to Life

Once the y-intercept has been plotted, the journey of bringing a line to life continues by plotting additional points based on the slope. Remember that the slope tells us how to move from one point to the next.

If we have a line with equation \(y = 3x + 2\), after plotting the y-intercept \((0, 2)\), we can plot more points by following the slope. With a slope of 3, or \(3/1\), it means we rise 3 units up for every 1 unit we go right on the graph.

Plotting multiple points helps ensure that your line is accurate. Once you have several points determined by using the slope, draw a line through them with a ruler to ensure precision.

• Start at the y-intercept.
• Use the slope to move vertically and horizontally from that point.
• Plot the new point.
• Repeat to find more points.
• Connect the dots with a straight edge for the final graph.

Following these steps, your line will emerge, showing the relationship between the variables as illustrated by the equation.