What are the roots of the given polynomial? What do double roots look like on a graph?
Practice makes perfect
We want to sketch the graph of the given function.
y = - x^2(x-2)^2(x+2)^2
Let's start by determining the degree of the function. The are 3 parentheses being multiplied. Let's take a look at the greatest power of x in each set of parentheses.
y = - x^2(x-2)^2(x+2)^2
If we were to multiply the highest powers of x in each set of parentheses, we would multiply the power of 2 three times. Remember that when we multiply powers with the same base, we add the exponents.
y= - x^2(x-2)^2(x+2)^2 ⇓
Degree: 2 + 2 + 2 = 6
This means that the degree of y is 6. Now we want to graph y. To do so, we will find x-intercepts and determine the function's end behavior.
Intercepts
The x-intercepts are the roots of the function — they are the solutions when y = 0. Let's use the Zero Product Property to find the roots of y.
- x^2(x-2)^2(x+2)^2 = 0 ⇕
x = 0 or x-2 = 0 or x+2 = 0
This gives us three roots: x = 0, x = 2, and x = - 2. Let's find the multiplicity of each root, or the number of times a zero can be factored from the polynomial. The multiplicity of the roots will help us graph the function.
Root
Multiplicity
0
2
2
2
- 2
2
In our case, every root is a double root.
Determining the End's Behavior and Sketching the Function
Let's determine the end behavior of y. The leading coefficient of the function is - 1 and the degree is 6.
y= - x^2(x-2)^2(x+2)^2 ⇓
Degree: 2 + 2 + 2 = 6
Since the degree is even and the leading coefficient is negative, the end behavior is down and down. Let's sketch the intercepts and the end behavior in the coordinate plane.
We are ready to sketch the graph! Remember that if a root has even multiplicity, the graph touches the x-axis at this point but does not cross it. If a root has odd multiplicity, the graph passes through it. In our exercise, all roots are double, so they are all even.
