Problem 53 - Brief Applied Calculus

Question

a. Graph the lines \(y_{1}=-x, y_{2}=-2 x\), and \(y_{3}=-3 x\) on the window \([-5,5]\) by \([-5,5]\). Observe how the coefficient of \(x\) changes the slope of the line. b. Predict how the line \(y=-9 x\) would look, and then check your prediction by graphing it.

Step-by-Step Solution

Answer

The line \( y = -9x \) is much steeper than the others.

Step 1: Understanding line equations

The general form of a linear equation is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Here, each line has no y-intercept (\( b = 0 \)) and is in the form of \( y = mx \).

Step 2: Identify slopes

The coefficients of \( x \) in the given equations are \(-1\), \(-2\), and \(-3\). These represent the slopes of the lines \( y_1 = -x \), \( y_2 = -2x \), and \( y_3 = -3x \), respectively.

Step 3: Graph first line \( y_1 = -x \)

Plot the line \( y = -x \) which has a slope of \(-1\). This means for every unit increase in \( x \), \( y \) decreases by one unit. On the window \([-5,5]\) by \([-5,5]\), it passes through the origin (0,0) and extends to (-5,5) and (5,-5).

Step 4: Graph second line \( y_2 = -2x \)

Plot the line \( y = -2x \) with a slope of \(-2\). For each unit increase in \( x \), \( y \) decreases by two units. The line crosses through the origin and points such as (-2.5, 5) and (2.5, -5).

Step 5: Graph third line \( y_3 = -3x \)

Plot \( y = -3x \) with a slope of \(-3\). Here, each unit increase in \( x \) causes a 3-unit decrease in \( y \). The line passes through points like (-1.67, 5) and (1.67, -5).

Step 6: Observe changing slopes

Increasing the absolute value of the slope makes the line steeper. From \( y_1 = -x \) to \( y_3 = -3x \), each line is steeper than the previous, and all lines pass through the origin.

Step 7: Predict line \( y = -9x \)

Based on previous observations, \( y = -9x \) should be steeper than \( y_3 = -3x \). It will cross the origin, and for each unit increase in \( x \), \( y \) will drop by 9 units.

Step 8: Graph line \( y = -9x \) and verify

Graph \( y = -9x \) on the same window. The line crosses the origin and illustrates a sharper decline than the others, consistent with the prediction, illustrating extreme steepness.

Key Concepts

Slope of a Line

The slope of a line, represented by the letter "m" in the equation of the form \( y = mx + b \), is a crucial concept in understanding linear equations. Essentially, the slope indicates how steep a line is and the direction it goes. The slope represents the change in the y-value for every one-unit increase in the x-value. In simpler terms, it is the "rise over run."

• A positive slope means the line ascends as you move from left to right.
• A negative slope means the line descends as you move from left to right.
• A larger absolute value of the slope indicates a steeper line.
• If the slope is zero, the line is horizontal; if undefined, it's vertical.

For example, in the lines identified in the exercise, \(y_1 = -x\) has a slope of -1, meaning as x increases by 1, y decreases by 1. This trend continues with \( y_2 = -2x \), where the slope is -2, leading to a sharper decline. Observe these changes as the slope magnitude increases, seeing how they affect the steepness of the line.

Graphing Lines

Graphing lines involves plotting points that a linear equation represents and connecting them to form a line. For linear equations like \( y = mx + b \), the process is simplified given that these lines form straight lines on the graph.
To graph a line:

• First, identify key components such as the slope (m) and y-intercept (b).
• Plot the y-intercept on the graph where the line crosses the y-axis.
• Use the slope to find another point on the line. For every 1 unit increase in x, shift the point up or down depending on the slope value.

In the given exercise, graphing \( y_1 = -x \), \( y_2 = -2x \), and \( y_3 = -3x \) involves choosing points across the x-axis from -5 to 5 and using their respective slopes. Since all three equations have no y-intercept, each passes through the origin (0,0) making it simpler.

Equation of a Line

The equation of a line in its most common form is \( y = mx + b \). In this formula, "m" as discussed is the slope, and "b" is the y-intercept, which indicates where the line crosses the y-axis.It is easy to write the equation of a line if these two values are known.
The interpretation of this equation is to see how each variable affects the graph:

• The slope, m, dictates the angle or steepness of the line.
• The y-intercept, b, shows the starting point of the line on the y-axis.

During the exercise, lines \(y_1 = -x\), \(y_2 = -2x\), and \(y_3 = -3x\) all lack the y-intercept term (b=0), meaning they slope downwards at varying rates from the origin itself.Predictions for other line equations such as \(y = -9x\) follow the same pattern with increased steepness, emphasizing the relationship between the equation and its graphical representation.