Wize Grade 11 Mathematics Textbook > Polynomials Factoring and Graphs

Graphing Polynomials using Transformations

Transformations on Polynomials
Transformations
If the polynomial can be written in the form
f(x)=a(k(x−d))n+cf(x) = \colorOne{a}(\colorTwo{k}(x - \colorThree{d}))^n + \colorFour{c}f(x)=a(k(x−d))n+c

then we can graph the polynomial using transformations on the basic polynomial.

Example 1

Graph the following polynomials.

1.  f(x)=2(x−1)3f(x) = 2(x - 1)^3f(x)=2(x−1)3

The basic polynomial is y=x3y = x^3y=x3
The 1 on the inside will translate the graph right one unit
The  2 on the outside will stretch it vertically by a factor of 2

2.  g(x)=−(x+2)4+2g(x) = - (x+2)^4 +2g(x)=−(x+2)4+2

The basic polynomial is y=x4y = x^4y=x4
The 2 on the inside will translate the graph left 2 units
The -1 on the outside will reflect it over the x-axis
The +2 on the outside will translate the graph up 2 units

Wize Concept
It is important to know the basic shapes of polynomials before applying transformations to them.   

Example:  Graphing Polynomials using Transformations
The following graph comes from transforming the polynomial f(x)=x4f(x) = x^4f(x)=x4


1.  What transformations can you read from the graph.

It appears that the graph has been
translated right by 2 units
translated down by 3 units
it maybe stretched, but it is unclear by how much

2.  Write the equation of the transformed function.  Begin the equation with y=y =y=.

From the transformations we know that y=a(x−2)4−3y = a(x - 2)^4 - 3y=a(x−2)4−3

To find the value of aaa we can use information from the point (1,0)(1,0)(1,0)
This gives us that

0=a(1−2)4−33=a(−1)43=a\begin{aligned}
0 &= a(1-2)^4 - 3 \\
3 &= a(-1)^4 \\
3 &= a \\
\end{aligned}033​=a(1−2)4−3=a(−1)4=a​

To the equation from the graph is y=3(x−2)4−3y = 3(x-2)^4 -3y=3(x−2)4−3

Practice: Graphing Polynomials using Transformations
The following is a transformed graph of the function f(x)=x3f(x) = x^3f(x)=x3


Select the correct equation for the transformed graph.

A) y=(x−2)3−1y = (x -2)^3 - 1y=(x−2)3−1

B) y=(x+1)3−2y = (x+1)^3 - 2y=(x+1)3−2

C) y=(x−1)2−2y = (x - 1)^2 - 2y=(x−1)2−2

D) y=−(x+2)3−1y =- (x + 2)^3 - 1y=−(x+2)3−1

I don't know

Practice: Graphing Polynomials using Transformations
For the following polynomial there are several transformations

y=2(4(x−5))3+6y=2(4(x-5))^3 + 6y=2(4(x−5))3+6

1.  Identify the original polynomial that was transformed

2.  What numbers correspond with each transformation

1. The original polynomial is

y =

I don't know

Practice: Graphing Polynomials using Transformations
Write the equation which transforms the original function f(x)=x2f(x) = x^2f(x)=x2
with the following:
Translated right by 7 units
Has a vertical stretch by 9
Is reflected over the x-axis
Translated down by 3

y =

I don't know