Graphing Factored Polynomials Sketch the graph of the polynomial function. Make sure your graph

Graphing Factored Polynomials Sketch the graph of the...

Key Concepts

End Behavior

The end behavior of a polynomial describes how the function behaves as x approaches positive or negative infinity. This behavior is determined by the degree of the polynomial and its leading coefficient, helping to shape the overall direction of the graph as it extends towards the ends.

Multiplicity of Zeros

The multiplicity of a zero indicates how many times the corresponding factor appears in the polynomial. This concept affects the graph's behavior at the x-intercepts—zeros with odd multiplicity typically cross the axis, while those with even multiplicity tend to only touch the axis and turn around.

Y-intercept

The y-intercept is the point where the graph crosses the y-axis. It is found by evaluating the polynomial at x = 0. This point provides an important reference for plotting the graph and understanding the initial value of the function.

Factored Form of the Polynomial

Expressing a polynomial in its factored form immediately reveals its zeros, making it easier to analyze and graph the function. The factors indicate the critical points (roots) where the polynomial equals zero, which are essential in determining the graph’s intercepts.

Zeros and X-intercepts

The zeros of the polynomial are the values that make the function equal to zero. These correspond to the x-intercepts of the graph. Identifying these points is crucial for sketching the accurate location of the graph on the coordinate plane.

Transcript

00:02 So remember that when we're sketching polynomials, the first thing we have to do is figure out the x intercepts.

00:08 And so we know we have an x intercept of negative 2 here, as well as an x intercept of negative 1.

00:16 And lastly, an x intercept at 3 halves, at positive 3 halves.

00:23 So this x is equal to negative 2 has a multiplicity of 1, which means this will cross the x -axis.

00:30 This x is equal to negative 1 has a multiplicity of 2 which means we will only touch the x -axis and lastly this x is equal to 3 halves is as a multiplicity of 1 so we're going to again cross the x -axis so let's look at our end behavior here well when x approaches negative infinity we have a negative infinity times a negative infinity which is positive infinity and then we have a negative infinity squared so that's also positive and so we have a positive times a positive so this gives us that our polynomial or our graph will approach positive infinity and so now let's see when we have x is positive infinity well when x is positive infinity we have a positive times a positive so this is a positive and we have another positive squared so positive times positive again is is a positive infinity so we have that our polynomial again approaches positive infinity so now let's catch this so we know that this is our y -axis this is our x -axis and we have we have x intercepts at at negative 2 negative 1 so let's say this is negative 1 this is negative 2 and if this is this is 1 and this is 2 then 3 halves will be right here 3 halves will be somewhere right in the middle so this is 3 halves this is negative 1 and this is negative 2 this is negative 2 so let's draw these points so we know we have an x intercept here, here, and here.

02:10 So let's see when x is zero, what is our y value? so when x is zero, we have positive two.

02:19 Let's write this up here somewhere.

02:21 When x is equal to zero, we have positive two times 1 times negative 3.

02:31 So this gives us negative 6...

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