PC - Rates of Change and Linear Equations Lesson

Rates of Change and Linear Equations

A rate of change expresses a relationship between two variable quantities. Consider a situation where every 3500 calories consumed over your daily requirement equates to a pound gained . Then the rate of change is 3500 calories/pound. As the number of non-required calories consumed increases by 3500, then the number of pounds gained also increases by 1. Consuming an extra 7000 calories nets a gain of 2 pounds. Likewise, consuming an extra 10,500 calories produces a 3 pound gain. This example illustrates what is considered a constant rate of change, and the plot of these three points suggests a linear relationship exists.

Slope of a Line and Rate of Change

An important foundational concept from your previous mathematical studies is slope. Conceptually, the slope of a line is interpreted as the steepness of a line or the rate at which a line rises or falls. The rate of change of a line is always constant, unlike a curve that has a variable rate of change. The slope of a line is calculated from changes in coordinates of any two points on the line and measures the number of units of vertical change (rise) for each unit of horizontal change (run).

Recall the familiar definition for the slope m of a non-vertical line passing through two points

(x₁,y₁) and (x₂,y₂) on the line:

 $m=\frac{rise}{run}=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}$

View the video below on computing the slope of a line.

Before continuing, explore how the slope of a line affects the line's orientation in the coordinate plane by viewing the Slope of a Line interactive tutorial listed in the More Resources area of the sidebar. Knowing the slope of a line enables us to draw conclusions about how the line is oriented in the coordinate plane. Click HERE to view the video on slopes rise and fall. Links to an external site. 

You may remember slope as a ratio of two numbers with no units attached. This occurs when the units associated with the two numbers are the same. One of the fundamental notions in calculus is interpreting slope as a rate of change of one variable with respect to another involving different units, e.g., ft/sec or miles/hour.

Rates of change come in two varieties:

Average rate of change is a ratio comparing the change in the value of one quantity to the change in value of another quantity over a closed interval (associated with the definition of slope).

Instantaneous rate of change is the rate of change at one specific instant/location/value (associated with the definition of derivative and defined more thoroughly in a subsequent module).

Slope-Intercept, Point-Slope, and Standard/General Equations of a Line

In order to write an equation for a line that is not vertical, it is sufficient to know one of the following: a) two points on the line, or b) one point on the line and the slope of the line.

Slope-Intercept Equation of a Line, y = mx + b, where m represents the slope of the line and b is the y-intercept, is very popular for graphing and for writing an equation of a line if the y-intercept is known.

Point-Slope Equation of a Line, y - y₁ = m(x - x₁) or alternatively y = m(x - x₁) + y₁, where m represents the slope of the line that passes through (x₁,y₁), is the most versatile form for writing an equation of a line since it uses any point on the line rather than the y-intercept. View the video below that demonstrates how to use the point-slope form of an equation to write the equation of a line.

Click HERE to explore the presentation below Links to an external site. to examine the behaviors of non-vertical/non-horizontal, horizontal, and vertical lines that may help you understand the reasoning behind how to write the equations of horizontal and vertical lines.

Every line no matter its orientation can be written using the Standard/General Equation of a Line, Ax + By = C, where A and B 0 and A, B, and C are integers. Some texts use Ax + By + C = 0 as the standard/general form of the equation of a line. Both are widely accepted as correct; it really just depends on which text you are using and the preference of your teacher. View the video on finding the equation of a line in standard form.

Graph of a Linear Equation in Slope-Intercept Form

Although graphing a linear equation given in slope-intercept form (y = mx + b) is perhaps very familiar to you, a review of the basics may be in order. When beginning to graph in the coordinate plane, the first skill you learn is plotting points. So it is with graphing a linear equation in slope-intercept form; you first plot a point. In this case the point is located on the y-axis and has coordinates (0, b). Once the point (0, b) is plotted, then another point must be found in order to connect these two points to form a line. Finding this second point is accomplished by using the given slope, which must be expressed as the ratio m/1 per the definition of slope, to move m units vertically (up if m is positive and down if m is negative) and 1 unit in a positive horizontal direction from (0, b). View the video illustrating how to graph equations written in slope-intercept form.

Equations of Parallel or Perpendicular Lines to a Given Line

The slope of a line is used to determine whether or not lines are parallel or perpendicular. Intuitively, parallel lines never intersect and always remain the same distance apart.

View the parallel lines video below.

Unlike parallel lines, perpendicular lines intersect to form four right angles at the point of intersection. There can be multiple lines perpendicular to a given line; however, given a point through which the perpendicular line must pass, only one perpendicular line is possible.

View the perpendicular lines video below.

Let's explore the relationships between the slopes of parallel and perpendicular lines.

• Two non-vertical lines are parallel if and only if they have the same slope.
• Two non-vertical lines with slopes $m_1$ and $m_2$ are perpendicular if and only if $m_1m_2=-1$. i.e., their slopes are negative reciprocals: 

$m_1=-\frac{1}{m_2}$

• Recall: The phrase "if and only if" is used when two implications are combined into one statement. "If and only if" is often abbreviated using "iff".
• "Two non-vertical lines are parallel if and only if they have the same slope" is interpreted as:
• If two non-vertical lines are parallel, then they have the same slope.
• If two non-vertical lines have the same slope, then they are parallel.
• "Two non-vertical lines with slopes $m_1$ and $m_2$ are perpendicular iff $m_1m_2=-1$" is interpreted as:
• If two non-vertical lines with slopes $m_1$ and $m_2$ are perpendicular, then $m_1m_2=-1$.
• If the product of the slopes of two non-vertical lines $m_1m_2=-1$, the lines are perpendicular.

Two proofs related to the slopes of parallel and perpendicular lines, as well as additional review related to parallel and perpendicular lines may be accessed from the More Resources section of the sidebar. Consider three examples that illustrate typical problems related to parallel and perpendicular lines that you've probably encountered in previous mathematics courses. View the examples below.

Click HERE to view Example 1 Links to an external site.

Click HERE to view Example 2 Links to an external site.

Click HERE to view Example 3 Links to an external site.

Finding an equation of a line passing through a given point and parallel or perpendicular to a given line requires an appropriate slope value and coordinates of a given point. Consider the following two situations below.

Rates of Change and Linear Equations Practice

Rates of Change and Linear Equations: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of content related to rates of change and linear equations.

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