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
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 to 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
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: Graph the line illustrating speed
To graph this line, we need the
Distance and time graph
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
Key Term Reference
 average
 Appears in these related concepts: Mean: The Average, Average Value of a Function, and Averages
 constant
 Appears in these related concepts: Graphing Quadratic Equations in Vertex Form, Inverse Variation, and Direct Variation
 denominator
 Appears in these related concepts: Solving Problems with Rational Functions, Introduction to Rational Functions, and Fractions Involving Radicals
 dependent variable
 Appears in these related concepts: Converting between Exponential and Logarithmic Equations, The Cartesian System, and What is a Quadratic Function?
 distance
 Appears in these related concepts: Inequalities with Absolute Value, The Distance Formula and Midpoints of Segments, and Linear Mathematical Models
 domain
 Appears in these related concepts: Restricting Domains to Find Inverses, Visualizing Domain and Range, and Composition of Functions and Decomposing a Function
 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: Functions and Their Notation, The Vertical Line Test, and Solving Differential Equations
 graph
 Appears in these related concepts: Graphical Representations of Functions, Graphing Equations, and Reading Points on a Graph
 linear
 Appears in these related concepts: Exponential Growth and Decay, Graphs of Linear Inequalities, and Factoring General Quadratics
 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, Polynomial and Rational Functions as Models, and Graphing Quadratic Equations In Standard Form
 set
 Appears in these related concepts: Sequences, Introduction to Sequences, and Sequences of Mathematical Statements
 variable
 Appears in these related concepts: Fundamentals of Statistics, Math Review, and Introduction to Variables
 velocity
 Appears in these related concepts: Velocity of Blood Flow, RootMeanSquare Speed, and Rolling Without Slipping
Sources
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