Concept Byte: Graphing Polynomials Using Zeros
Sign In
Sketch:
Graphing calculator:
To sketch the graph of a function, we must find its zeros and determine how the function changes around these points.
Write as a power
Split into factors
a^2-2ab+b^2=(a-b)^2
Use the Zero Product Property
(I), (II): sqrt(LHS)=sqrt(RHS)
Having fully factored the function, we see that x=2 and x=- 2 are the only rational roots. Let's plot them in a coordinate plane.
If we know how the functions behaves around these zeros, we get an idea of what it looks like. To determine this, we can calculate the f-values in the following intervals. x&<- 2 - 2< x& < 2 2< x& Let's choose some x-values in these intervals and find their corresponding f-values. The only x-value that will not be arbitrary is x=0, as we want to know where the graph intercepts the y-axis.
| Interval | x | (x+2)^2(x-2)^2 | f |
|---|---|---|---|
| x<- 2 | - 3 | ( - 3+2)^2( - 3-2)^2 | 25 |
| - 2< x < 2 | 0 | ( 0+2)^2( 0-2)^2 | 16 |
| 2 < x | 3 | ( 3+2)^2( 3-2)^2 | 25 |
With the exception of the y-intercept, the actual f-values for the given x-values we used are not important. Instead, we are more interested if the function is above or below the x-axis in the given intervals. This will tell us how the function behaves. f(- 3)&= 25 &&⇒ Abovethex-axis f(0)&= 16 &&⇒ Abovethex-axis f(3)& = 25 &&⇒ Abovethex-axis The graph is above the x-axis before x=- 2 and above the x-axis after x=- 2, so the graph is decreasing before the zero and increasing after it. By the same reasoning, the graph is decreasing before x=2 and increasing after. Notice that we will also draw our graph so it intercepts the y-axis at (0,16) for a more accurate sketch.
The graphing calculator image is very similar to our graph.