Lesson 2.7: Building Functions

Lesson 2.7.1: Building Functions02/27/2015

Comparing Linear & Exponential FunctionsLinear ModelExponential ModelGeneral Form f(x)=mx+b f(x)= a

Meaning of parameters m is the slope of the line or the constant rate of changeb is the y-intercepta is the initial value or y-intercept.b is the base or constant factor of changeDifference between consecutive y-valuesThere is a constant difference between consecutive y-valuesThere a constant factor between the difference in the consecutive y-valuesExample

Difference between consecutive y-valuesXF(x)03133236333993

XG(x)0321212103-1

303303630The 1st difference between consecutive y-values is not constant SO this function is not linear but exponential [f(x) = ab^x ] -11

-11

-11The 1st difference between consecutive y-values is constant SO this is a linear function [f(x) = mx + b]

3

1.- Analyzing the consecutive 1st differences between y-values: 21-32=-11; 10-21=-11; 10-(-1)=-11; -12-(-1)= -11 } we can see that there is a constant difference (-11) between consecutive y-values which tell us that the table is representing a linear function of the form f(x) = mx + b we can eliminate option (d) since that is an exponential function [f(x) = ab^x].

2.- We need to find the slope of the line, a.k.a, the rate of change which is -11!!!!

3.- We could go all the way and find the y-intercept [given by the point (0,y), but since we know the the type of function (linear) and the slope of the line (-11) we can determine that, so far, our linear function looks like f(x) = -11x +b , SO option (c) is the only alternative that reflects what we found (in this case the slope of the line was the same as the 1st difference only because the x-values are increasing by 1 but in general you have to use the slope equation, m= y2 y1 / x2 x1 )

Year (x)Value (y)0200,0001220,0002240,0003260,000

1.- The difference between consecutive values of y is constant (20,000) so we are in the presence of a linear function [f(x)=mx+b] so we can eliminate options (c) and (d) since they are of the form f(x)=ab^x which are exponential functions.

2.- the slope of the line, m= y2-y1/x2-x1 = 20,000.

3.- the y-intercept is where we have the point (0,y) when x=0, y=200,000! The initial value of the house.

4.- so, our function will be f(x)= 20,000x+200,000, which is option (b)

1.- Obviously here we are in the presence of a linear function [f(x)=mx+b] eliminate option c!

2.- We could calculate the slope of the line but lets get smart. The function presents a negative slope (-m) which reduces our our options to either (a) or (d)

3.- We can clearly see in the graph that the y-axis gets crossed at (0,3), y-intercept = 3

4.- Option (a) is the one that has a negative slope and a positive 3 as y-intercept

ANWERS:

1.- We will be ask to find one value with respect to another value. In general we talk about the x-variable as the independent variable (like days, weeks, months, years, or number of figures, etc), and the y-variable as the dependent one (this one will increase or decrease with respect of the days, weeks, months, years, or numbers of figures )

2.- The slope or rate of change of an exponential function (called rate of grow when the slope is +, and rate of decay when the slope is -), is faster (or bigger) than the rate of change of a linear function. An exponential function will increase or decrease faster than the increase or decrease of a linear function.

3.- Check the next slide!!

From the web:

Building Functions from Context

1.- Identify the independent and dependent quantities2.- Make a table3.- Find the type of difference between consecutive y-values4.- Determine type of function5.- Write the function

Example 1

1.- Identify the independent and dependent quantities: months!! The months will pass independently of the money she has in her account.

2.- Make a table:

She started with this amount!

3.- Find the type of difference between consecutive y-values: Easy, 30! The problem says that she takes $30.00 each month!!4.- Determine type of function: since the 1st difference between consecutive y-values is constant ($30.00), then the function is linear!5.- Write the function: f(x) = mx+b f(x) = 30x+1250

months$0125011220 = 1250-3021190 = 1220 -3031160 = 1190 - 30

1.- Identify the independent and dependent quantities2.- Make a table3.- Find the type of difference between consecutive y-values4.- Determine type of function5.- Write the function

Example 21.- Identify the independent and dependent quantities: the number of figures is the independent quantity and the number of angles will be the dependent one2.- Make a table

3.- Find the type of difference between consecutive y-values: 4-2=2; 8-4=4; 16-8=8.4.- Determine type of function: since the 1st difference between consecutive y-values is not constant but increases then the function is not linear but exponential [f(x)=ab^x]5.- Write the function: analyzing the 1st differences I can see that each difference was multiplied by 2, 2x2=4, 4x2=8, so the factor of increase is 2 b=2. Now a=y-intercept which can be solved from the function: f(x)=ab^x; to solve for a we have to isolate it! So we divide both sides of the equation by b^x f(x)/b^x = a; for x=1, y=2, and b=2then a= 2/2^1 = 1!

Finally, the function is f(x)= ab^x = (1)(2^x)

Figure (x)# of angles (y)122438416

1.- Identify the independent and dependent quantities2.- Make a table3.- Find the type of difference between consecutive y-values4.- Determine type of function5.- Write the functionSteps to build a function from scratch!