February 24, 2016

Linear Functions

 

The first administration of the new SAT is right around the corner! If you took the PSAT last semester, you have had your first glance at the new material. If you have access to a practice PSAT, this is a great study resource. If not, Khan Academy (khanacademy.org) has plenty of practice problems to help you begin.

Types of Relationships

As you begin your practice, you will start to notice that the new SAT focuses on three types of numerical relationships: linear, quadratic, and exponential. Let’s look at some key features of linear functions this month.

Linear Functions

A linear function can be recognized by an arithmetic, consistent change in the data. If looking at a table of values, the y-values (or output) will have a constant change between them. This change will be either addition or subtraction.

Example:

x -1 0 1 2 4 7
y 30 25 20 15 5 -10

 

Notice that as x increases by 1, the y value decreases by 5. It follows then that if x increases by 2, the y value decreases by 10. This relationship is consistent for the entire table, so this is a linear relationship with a slope of -5.

In a word problem, linear relationships are needed when adding and subtracting by a constant value.

Often, linear word problems will give an equation and ask the reader to interpret the meaning of the slope or y-intercept of the equation. Slope can always be described as the change in y over the change in x. The y-intercept can be described as the initial value or the value when x = 0.

Example:

The cost in dollars to produce x number of shoes at a factory can be modeled by the following equation: C = 3/2 x + 75. Interpret the meaning of the number 75 in context to the problem. Interpret the meaning of the number 3/2 in context to the problem.

The 75 would represent that it costs the company $75 to produce 0 shoes. The 3/2 is slightly trickier. First, identify what the units of x and y are. We can see that C (cost) is in the usual position of y and is in dollars, and x is in units of shoes. This means that as the cost increases by $3, the number of shoes produced increases by 2 shoes. To phrase this better, it costs the company 3 more dollars to produce 2 more shoes. We can also say that it costs the company $1.50 more to produce 1 more shoe.

 

Another possibility for a linear word problem is to describe an equation and have the student create the equation.

Example:

In 2014, County X had 783 miles of paved roads. Starting in 2015, the county has been building 8 miles of new paved roads each year. At this rate, if n is the number of years after 2014, write a function f which gives the number of miles of paved road there will be in County X.

 

We can see that this is a linear relationship because of the consistent addition of 8 roads per year. Since this is the rate, this is the slope of the line. Next, we need to find the initial amount (y-intercept). County X starts with 783, so this is the initial amount. We can now plug these values into the y = mx + b equation for a line: y = 8x + 783. Now, let’s change the variables: n represents years (x value) and f is the name of function (this represents y). The final answer would be: f(n) = 8n + 783.

Next month, we will look more closely at exponential functions. To identify these, look for consistent multiplication/division instead of addition/subtraction.

Test Yourself!

A voter registration drive was held in Town Y. The number of voters, V, registered T days after the drive began can be estimated by the equation V = 3,450 + 65T. What is the best interpretation of the number 65 in this equation?

  1. The number of registered voters at the beginning of the registration drive
  2. The number of registered voters at the end of the registration drive
  3. The total number of voters registered during the drive
  4. The number of voters registered each day during the drive

 

 

 

 

 

Correct Answer: D

 

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