Exercise 15 Page 592

We will start by determining the type of function that best models the data. Then we will be able to write an exact equation that models the data.

Finding a Model

We want to determine the most appropriate model for the given data set. Note that the x-values have a common difference of 1. Therefore, we can check if the y-values have a common difference, a common ratio, or constant second differences. It will tell us which model is most appropriate for the data set.

The y-values have:	The model is:
A common difference	Linear
A common ratio	Exponential
Constant second differences	Quadratic

Sometimes data does not fall exactly into the models listed above. In that case, we have to round our results and see when they are the closest to being constant. Let's take a look at our table.

x	y
0	0
1	3
2	11.3
3	24.7
4	43.3

We can exclude the possibility of an exponential model, since the y-value in the first pair is 0. Let's check for a common difference!

x	y	First differences
0	0
1	3	+3 ↩
2	11.3	+8.3 ↩
3	24.7	+13.4 ↩
4	43.3	+18.6 ↩

As we can see, the differences between the consecutive y-values are far from being constant. Therefore we have to check the second differences.

x	y	First differences	Second differences
0	0
1	3	+3 ↩
2	11.3	+8.3 ↩	+5.3 ↩
3	24.7	+13.4 ↩	+5.1 ↩
4	43.3	+18.6 ↩	+5.2 ↩

The second differences of the y-values are all approximately 5, so a quadratic model fits the data.

Writing an Equation

We know that a quadratic function best models the given data. y=ax^2+ bx+ c To write an equation that models the data, we have to determine the values of a, b, and c. Recall that (0, c) is the y-intercept of the quadratic function. Since the pair (0,0) is included in the data set, c= 0. y=ax^2+ bx+ 0 ⇔ y=ax^2+ bx In order to determine the values of a and b, we will use two (x,y) pairs from the data set, other than (0,0), to write a system of equations. We will use (1,3) and (2,11.3).

y=ax^2+bx

SubstituteII

x= 1, y= 3

3=a( 1)^2+b( 1)

▼

Simplify

CalcPow

Calculate power

3=a(1)+b(1)

Multiply

Multiply

3=a+b

RearrangeEqn

Rearrange equation

a+b=3

This will be the first equation of our system. We will write the second equation using the point (2,11.3).

y=ax^2+bx

SubstituteII

x= 2, y= 11.3

11.3=a( 2)^2+b( 2)

▼

Simplify

CalcPow

Calculate power

11.3=a(4)+b(2)

Multiply

Multiply

11.3=4a+2b

RearrangeEqn

Rearrange equation

4a+2b=11.3

Let's write our system of equations. a+b=3 & (I) 4a+2b=11.3 & (II) We will use the Substitution Method. Let's start by solving the first equation for b. Then we will be able to substitute the expression equivalent to b into the second equation.

a+b=3 4a+2b=11.3

▼

Solve by substitution

SubEqn

(I):LHS-a=RHS-a

b=3-a 4a+2b=11.3

Substitute

(II):b= 3-a

b=3-a 4a+2( 3-a)=11.3

Distr

(II):Distribute 2

b=3-a 4a+6-2a=11.3

SubTerm

(II):Subtract term

b=3-a 2a+6=11.3

SubEqn

(II):LHS-6=RHS-6

b=3-a 2a=5.3

DivEqn

(II):.LHS /2.=.RHS /2.

b=3-a a=2.65

RoundInt

(II):Round to nearest integer

b=3-a a≈ 3

(I):a ≈ 3

b≈ 3- 3 a≈ 3

SubTerm

(I):Subtract term

b≈ a≈3

We have that a≈3 and b≈ 0. Let's finish writing the equation to model the data! y=3x^2+ 0x ⇔ y=3x^2 Below we have included a graph that shows how the equation models the given data.