We want to graph the given polynomial function.
f(x)=1/2(x+1)(x-4)^2
Let's start by finding its zeros.
Zeros of the Function
To do so, we need to find the values of x for which f(x)=0.
f(x)=0 ⇔ 1/2(x+1)(x-4)^2=0
Since the function is already written in factored form, we will use the Zero Product Property. Notice, we do not set the coefficient 12 equal to zero as it produces a false statement.
We found that the zeros of the function are - 1 and 4.
Graph
To draw the graph of the function, we will find some additional points and consider the end behavior. Let's use a table to find additional points.
x
12(x+1)(x-4)^2
f(x)= 12(x+1)(x-4)^2
- 12
12(( - 12)+1)(( - 12)-4)^2
5.063
0
1/2( 0+1)( 0-4)^2
8
2
1/2( 2+1)( 2-4)^2
6
3
1/2( 3+1)( 3-4)^2
2
The points ( 12, 5.063), ( 0, 8), ( 2, 6), and ( 3, 2) are on the graph of the function. Finally, let's apply the Distributive Property. This will simplify the equation and determine the leading coefficient and degree of the polynomial function.
We can see now that the leading coefficient is 12, which is a positive number. Also, the degree is 3, which is an odd number. Therefore, the end behavior is down and up. Now, let's draw the graph!
