Problem 12 - Technical Mathematics with Calculus

Question

Find the slope and \(y\) intercept of each straight line and make a graph. $$y=-1.75 x-5.44$$

Step-by-Step Solution

Answer

The slope of the line is -1.75, and the y-intercept is at \(y = -5.44\). The graph is a straight line passing through \((0, -5.44)\) and sloping downwards as you move to the right.

Step 1: Identify the Slope and Y-Intercept

The equation given is in the slope-intercept form, which is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. For the equation \(y=-1.75 x-5.44\), the slope \(m\) is -1.75 and the y-intercept \(b\) is -5.44.

Step 2: Graphing the Y-Intercept

Start by plotting the y-intercept on the graph. The y-intercept is the point where the line crosses the y-axis. Plot the point \((0, -5.44)\) on the graph.

Step 3: Plotting Another Point Using the Slope

To plot another point, use the slope \(-1.75\). Starting from the y-intercept, move 1 unit to the right (positive x-direction), and then 1.75 units down (negative y-direction) since the slope is negative. This leads to the point \((1, -7.19)\).

Step 4: Drawing the Line

Using the y-intercept and the second point, draw a straight line through these points. This line represents the equation \(y = -1.75x - 5.44\).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way to express a linear equation so that you can easily identify the slope, which shows the steepness of the line, and the y-intercept, the point where the line crosses the y-axis. This form is written as y = mx + b, where m is the slope and b is the y-intercept. It's a favorite among students and teachers for graphing because it makes the process straightforward.

For instance, in the equation y = -1.75x - 5.44, the slope-intercept form quickly tells us the slope m is -1.75, which means the line slopes downwards from left to right, and the y-intercept b is -5.44, indicating the line crosses the y-axis below the origin.

Y-Intercept

The y-intercept of a line is essentially the starting point on a graph. It's where your linear equation meets the y-axis, and it's represented by the coordinate (0, b). This value is crucial because it provides a reference point for graphing the entire line.

In our example, the y-intercept b is -5.44. To graph it, you locate the point (0, -5.44) on the y-axis. This is the first point you plot before using the slope to find other points on the line. Remember, no matter how a line tilts or where it goes, it will always pass through its y-intercept, making it an anchor for graphing.

Slope of a Line

The slope is a number that tells us how a line inclines or declines on a graph. We often think of slope as 'rise over run', where 'rise' refers to the vertical change and 'run' to the horizontal change between two points on the line. If the slope is positive, the line rises as you move from left to right. Conversely, if the slope is negative, the line falls as you move from left to right.

For our equation, the slope m is -1.75, a negative value. This means for every 1 unit you move right along the x-axis, you move 1.75 units down. To put this into action from the y-intercept, if you move 1 unit to the right you should then move 1.75 units down to find a new point on the line, such as (1, -7.19). Connecting this point with the y-intercept allows for the accurate drawing of the line on the graph.

Understanding Slope in Real Life

Think of slope as a hill's steepness. The greater the slope, the steeper the hill. A negative slope means you’re going downhill, while a positive slope means you’re climbing up.