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Rates of Change
Linear functions apply to real world problems that involve a constant rate.
Learning Objective

Apply linear equations to solve problems about rates of change
Key Points
 If you know a realworld problem is linear, such as the distance you travel when you go for a jog, you can graph the function and make some assumptions with only two points.
 The slope of a function is the same as the rate of change for the dependent variable
$(y)$ . For instance, if you're graphing distance vs. time, then the slope is how fast your distance is changing with time, or in other words, your velocity.
Terms

slope
The ratio of the vertical and horizontal distances between two points on a line; zero if the line is horizontal, undefined if it is vertical.

rate of change
Ratio between two related quantities that are changing.

linear equation
A polynomial equation of the first degree (such as
$x=2y7$ ).
Full Text
Introduction to Rate of Change
Linear equations often include a rate of change. For example, the rate at which distance changes over time is called velocity. If two points in time and the total distance traveled is known the rate of change, also known as slope, can be determined. From this information, a linear equation can be written and then predictions can be made from the equation of the line.
If the unit or quantity in respect of which something is changing is not specified, usually the rate is per unit of time. The most common type of rate is "per unit of time", such as speed, heart rate and flux. Ratios that have a nontime denominator include exchange rates, literacy rates and electric field (in volts/meter).
In describing the units of a rate, the word "per" is used to separate the units of the two measurements used to calculate the rate (for example a heart rate is expressed "beats per minute").
Rate of Change: Real World Application
Example 1: An athlete begins he normal practice for the next marathon during the evening. At 6:00 pm he starts to run and leaves his home. At 7:30 pm, the athlete finishes the run at home and has run a total of 7.5 miles. How fast was his average speed over the course of the run?
The rate of change is the speed of his run; distance over time. Therefore, the two variables are time
Example 2: Graph the line illustrating he speed.
To graph this line, we need the
Distance Time
The graph of
With this new function, we can now answer some more questions.

How many miles did he run after the first half hour? Using the equation, if
$x=\frac{1}{2}$ , solve for$y$ . If$y=5x$ , then$y=5(0.5)=2.5$ miles.  If he kept running at the same pace for a total of
$3$ hours, how many miles will he have run? If$x=3$ , solve for$y$ . If$y=5x$ , then$y=5(3)=15$ miles.
There are many such applications for linear equations. Anything that involves a constant rate of change can be nicely represented with a line with the slope. Indeed, so long as you have just two points, if you know the function is linear, you can graph it and begin asking questions! Just make sure what you're asking and graphing makes sense. For instance, in the marathon example, the domain is really only
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Key Term Reference
 constant
 Appears in these related concepts: Inverse Variation, Graphing Quadratic Equations in Vertex Form, and Direct Variation
 denominator
 Appears in these related concepts: Complex Conjugates and Division, Rational Equations, and Fractions Involving Radicals
 dependent variable
 Appears in these related concepts: The Cartesian System, What is a Quadratic Function?, and Converting between Exponential and Logarithmic Equations
 distance
 Appears in these related concepts: Inequalities with Absolute Value, Linear Mathematical Models , and The Distance Formula and Midpoints of Segments
 domain
 Appears in these related concepts: Composition of Functions and Decomposing a Function, Visualizing Domain and Range, and Restricting Domains to Find Inverses
 equation
 Appears in these related concepts: Equations and Inequalities, Graphs of Equations as Graphs of Solutions, and What is an Equation?
 function
 Appears in these related concepts: Introduction to Domain and Range, Solving Differential Equations, and The Vertical Line Test
 graph
 Appears in these related concepts: Reading Points on a Graph, Graphing Equations, and Graphical Representations of Functions
 linear
 Appears in these related concepts: Factoring General Quadratics, Exponential Growth and Decay, and Graphs of Linear Inequalities
 linear function
 Appears in these related concepts: Zeroes of Linear Functions, What is a Linear Function?, and Increasing, Decreasing, and Constant Functions
 point
 Appears in these related concepts: The Intermediate Value Theorem, Graphing Quadratic Equations In Standard Form, and Polynomial and Rational Functions as Models
 set
 Appears in these related concepts: Sequences of Mathematical Statements, Sequences, and Introduction to Sequences
 variable
 Appears in these related concepts: Fundamentals of Statistics, Math Review, and Introduction to Variables
 velocity
 Appears in these related concepts: Tangent and Velocity Problems, Force, and Average Velocity: A Graphical Interpretation
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Source: Boundless. “Rates of Change.” Boundless Algebra. Boundless, 06 Jun. 2016. Retrieved 29 Jul. 2016 from https://www.boundless.com/algebra/textbooks/boundlessalgebratextbook/linearfunctions5/applicationsoflinearfunctions37/ratesofchange1035009/