2.4: Families of Polynomial Functions

Hello fellow students, I’m sorry for the delayed post for Ms. Burchat’s third period class.

My scribe post was Friday’s lesson, lesson 2.4: Families of Polynomial Functions. A good way to think of families of polynomial functions is like flowers. How flowers you say? Well think of it this way, if you choose one type of flower; for instance a rose, all roses are typically the same making them a family. However what differentiates between them are their colours and sizes, this would specify or identify a specific rose from the family of roses.

So how do roses relate to functions? Now that you understand what is meant by a family you can understand families of functions. When we say families of polynomial functions we mean to say in a linear, quadratic, cubic, quartic or quintic function these would be know as your families and to differentiate from one quadratic to another it would depend on the numerical values of each quadratic function which would make that individual function a specific one from the family of quadratic functions.

In Friday’s class we learned to do so algebraically, here is an example.

If we say we have the zeros of a family for cubic functions are 4, -1, and 3

Then we say:
a) Determine an equation for this family.
b) Determine an equation for the member of the family whose graph has a y-intercept of 17.

Solution:
a) Since the zeros are: 4, -1, and 3. (x-4)(x+1)(x-3) are also factors of the family of cubics. Therefore the equation for this question would be simply just to equate the factors to 0 and solve for x, as follows:

x-4= 0, then x= 4
x+1=0, then x= -1
x-3=0, then x=3
So, now we write the equation:
y= (x-4)(x+1)(x-3)

b) Now, we sub in the y-intercept in order to find the value of K. Using the equation we created above: y=k (x-4)(x+1)(x-3), we will put instead of y, 17 and 0’s for the x- intercepts.

Make note that we are using 0’s for the x-intercept and not an actually value because in the question it did not give us a pair of points that passed through the function like follows: (5,17). In this case we would sub in X with 5 and y with 17. But since we are only given the y- intercept we assume x is 0.

So with the function: y= k(x-4)(x+1)(x-3)

17= k(0-4)(0+1)(0-3)
17= -4k+1-3
17=2k
17/ 2=k
8.5=k
And therefore, y= 8.5(0-4)(0+1)(0-3)
This would be a SPECIFIC cubic function from the many members of the family of cubics.