Problem 38 - Introductory and Intermediate Algebra for College Students 4th

Question

On the same set of axes, draw lines with \(y\) -intercept 4 and slopes \(-1,-\frac{1}{2}, 0, \frac{1}{3},\) and 2.

Step-by-Step Solution

Answer

Sketch the line graph for each of the given slopes starting from the y-intercept (0, 4). The steepness varies in each line as per their slopes.

Step 1: Understanding Slope and Y-intercept

The slope of a line is the measure of the vertical change (rise) for each unit of horizontal change (run). The y-intercept is the point where the line crosses the y-axis. In the equation of a straight line (when the equation is in the form \(y = mx + c\)), 'm' represents the slope and 'c' is the y-intercept.

Step 2: Drawing the line for slope -1

The line with a slope of -1 passes through the y-intercept (0, 4). Starting from the point (0, 4), for each step to the right on the x-axis, go one step down on the y-axis because the slope is -1.

Step 3: Drawing the line for slope -1/2

The line with a slope of -1/2 passes through the y-intercept (0, 4). Starting from the point (0, 4), for every 2 steps to the right on the x-axis, go one step down on the y-axis as the slope is -1/2.

Step 4: Drawing the line for slope 0

The line with a slope of 0 is a horizontal line that passes through the y-intercept (0, 4). So, draw a line parallel to x-axis passing through the point (0, 4).

Step 5: Drawing the line for slope 1/3

The line with a slope of 1/3 passes through the y-intercept (0, 4). Beginning at the point (0, 4), for every 3 steps to the right on the x-axis, go one step up on the y-axis as the slope is 1/3.

Step 6: Drawing the line for slope 2

Start from the y-intercept (0, 4). The line with a slope of 2 will go up 2 units on the y-axis for every one unit to the right on the x-axis. So, for every step to the right on the x-axis, go two steps up on the y-axis.

Key Concepts

Slope of a Line

The slope of a line is a number that describes both the direction and the steepness of the line. It is often denoted by the letter m. Imagine you're climbing a hill; the slope tells you how steep the hill is. The mathematical formula for calculating slope is the ratio of the change in the y-value (rise) to the change in the x-value (run). This gives us the formula:

\[\begin{equation} m = \frac{\text{rise}}{\text{run}} \end{equation}\]

Positive slopes mean the line goes uphill when moving from left to right, while negative slopes go downhill. A slope of zero indicates a flat, horizontal line. On the exercise provided, slopes like -1 and 2 indicate lines that are going downwards and upwards, respectively, as you move to the right along the x-axis.

Y-intercept

The y-intercept of a line is the point where the line crosses the y-axis on a graph. It’s where the line would