Matching and Graphing Polynomial Functions

Hello, I'm Alesandro.

This March 12th we learned about matching polynomial functions to their graphs and graphing polynomial functions.

First, you must identify various parts and characteristics of a polynomial function to be able to picture and graph it.
These characteristics include:

• Type of function, degree, and end behaviour


• A function with an even degree would start and end in the same direction  
• (x → -∞ ; f(x)→ + ∞ ) (x → + ∞ ; f(x)→ + ∞ )
• A function with an odd degree would rise and fall
• (x → -∞ ; f(x)→ - ∞ ) (x → + ∞ ; f(x)→ + ∞ )

• Possible x-intercepts ( y = 0 )
• Number of x-intercepts vary depending on degree
• Possible y-intercept ( x = 0 )
• Maximum or Minimum value

If you are tasked with matching graphs, the easiest method would be to determine the y-intercept. This is when x is given the value of 0 and the y-intercept would be the last term, or the term without the variable of x.

Now let's get to graphing.

• To be able to graph a polynomial function, use the x-intercepts, the y-intercept, the degree of the function, and the sign of the leading coefficient.
• x-intercepts are the roots of the corresponding polynomial equation (x-intercepts of a quintic function would be the roots of the quartic equation, and so on)
• The zeroes are factors of the polynomial function
• The factor theorem is used to express the factored form of a polynomial function

Steps:

1. Factor the equation
2. The sign of the leading coefficient should be stated and the end-behaviour arrows drawn.
• The sign of the leading coefficient determines the direction the end-behaviour arrows go to.
• The degree determines the placement of the end-behaviour arrows as well.
• An odd degree with a positive leading coefficient would have the left arrow go down (QIII) and the right would go up (QI)
• An odd degree with a negative leading coefficient would have the left arrow in QII and the right at QIV
• An even degree with a positive leading coefficient would have the arms both go up (QI and QII);
• An even degree with a negative leading coefficient will have both arrows go down (QIII and QIV)
3. Plot the zeros, the y-intercept, and connect them by using curved lines

Note: 

To determine the lowest or highest point (where the curve line will turn) between two points, you must substitute x for a value between the two given points.
   If the curve is between (2,0) and (4,0) the curve would be located between these two points such as (3,y)

Important Graphing Rules:

1. If the root of the equation is unique (not having two or more of the same root) the curve passes the x-axis at that point.

     2. If there are two or more of the same root:

• An even number of the same root would cause the curve to bounce off at the x-axis

             

• An odd number of the same root would cause the curve to cross at that point in the x-axis

    3. The higher the degree, the more the graph flattens out at this point.

That's all guys and thanks for your time!

Note²:  Lack of unit booklet images brought to you by a non-responding camera on a smartphone and a       malfunctioning scanner. Sorry.

Note³: If you have trouble imagining graphs and afraid of downloading software, a free graphing calculator is available online @ https://www.desmos.com/calculator