Introduction to rules for linear functions:
Linear functions are very simple functions in mathematics. It
represents the relationship between the input and output functions. In other words we can say that the change in the input will change the value of the output. So we can say that input is in
direct relation with the output function. In this input function is called independent function and the output function is called dependent function since it depend on the input function. Linear
function is very simple to write since there will constant increment in the output according to the corresponding input. Let us see the rules to be followed for solving linear
functions.
Rules for writing linear functions are as follows:
We can add the same number to both sides of a linear equation.
Example: Consider the equation: x - 3 = 7 Then, x - 3 = 7 x - 3 + 3 = 7 + 3 [adding 3 on both sides] x = 10 . . . x = 10 is solution of...
Identify the input and output variables.
Represent, examine, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules.
Relate and match up to different forms of representation for a relationship.
Identify the function as linear or nonlinear and contrast their properties from tables, graphs, or equations.
Use representative algebra to symbolize situations and to solve problems, especially those that involve linear relationships.
Recognize and produce equivalent forms for simple algebraic expressions and solve linear equations.
Model and solve contextualized problem using various representation, such as graphs, tables, and equations.
Use graphs to examine the nature of changes in quantities in linear relationships
A high school have 1200 students enrolls in 2003 and 1500 students in 2006. If the student population P; grow as a linear function of time t, where t is
the number of years after 2003.
a) How many students will be enrolls in the school in 2010?
b) Find a linear function that relates the student populations to the time t.
Solution:
a) The given in order may be written as ordered pairs (t , P). The year 2003 corresponds to t = 0 and the year 2006 corresponds to t = 3, hence the 2
ordered pairs
(0, 1200) and (3, 1500)
Since the population grows linearly with the time t, we use the two ordered pairs to find the slope m of the graph of P as follows
m = (1500 - 1200) / (6 - 3) = 100 students / year
The slope m = 100 means that the student population grow by 100 students every year. From 2003 to 2010 present are 7 years and the students populations in
2010 will be
P (2010) = P(2003) + 7 * 100 = 1200 + 700 = 1900 students.
b) We know the slope and two points, we may use the point slope form to find an equation for the population P as a function of t as
follows
P - P1 = m (t - t1)
P - 1200 = 100 (t - 0)
P = 100 t + 1200