Graphing Polynomial Functions
Introduction
I think I was hesitant to do this video. Don't know if I just was tired of polynomials altogether but definitely was not keen which is odd because I am a picture person so here goes. Before reviewing the youtube I thought I might grab some examples.
Even though I had been doing polynomials I have not graphed one during this time but looking at them, it became clearer what I am trying to understand. I did know that the degree determines the number of factors but it never occurred to me how that looked visually.As in two factors, cross the x-axis twice, 3 factors cross the x-axis 3 times etc
So the video starts off talking about the direction of the graph. So you can see negative down, positive up.
I guess this shows it better
And he went of to talk about the behavior of functions when the degree was odd and even and the leading coefficient is positive of negative. For instance A positive x² (even degree) is a parabola, where as a cube is an s-shaped graph.
Multiplicity
This was hard to grasp for me as it sounded like degree as the tutor went on to say m = 1, m = 2, m = 3. But multiplicity is about how often a factor repeats e.g. This is a 2 multiplicity.
f(x) = (x-1)(x-1)
This is a 4 multiplicity.
f(x) = (x+4)(x+4)(x+4)(x+4)
And they care about this because multiplicity tells you how the graph behaves at that root. A root is a number where the polynomial equals zero.
m = 1 (simple root)
- Graph crosses the x‑axis normally
- Looks like a straight line passing through
m = 2 (double root)
- Graph touches the x‑axis and bounces off
- Looks like a parabola at that point
m = 3 (triple root)
- Graph crosses, but flattens first
- Looks like an S‑curve through the axis
m = 4, 5, …
- Even m → touch and bounce
- Odd m → cross, but flatter as m increase
Here are the first 3 graphs demonstrating m 1-3
Interpreting Graphs
So you can see from the image below there is a lot you can derive from a graph.
A Example
Don't know why I am surprised at how many pictures I have had to do. It is about graphs. I really liked the next bit. We started with
y = (x+2)(x-1)²(x-4)
So now I see the light. As the multiplicity is for each factor. So we have m = 1, m = 2 and m = 1. He then proceed to find the zeros, So -2, 1 and +4 and plotted them. Looking at the function we can see that this is a 4 degree function so x⁴. This means, because it is even it will be up, up. Like a parabola. So drew both left and right from the top to the plotted zero at -2 and 4. Looking at point 1 for (x-1)², the multiplicity is 2 so it will be a x² line but upside down. Once he had the graph drawn he put real number into the expression to see how low the curve was. He put in -1 and 3 which gives
y = (-1+2)(-1 -1)²(-1-4) = 1 * - 4 * -5 = -20 y = (3+2)(3-1)²(3-4) = 5 * 4 * -1 = -20
So this is what it looks like
