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

Graphing Factored Polynomials Sketch the graph of the...

Key Concepts

Factored Polynomial

Understanding a polynomial presented in factored form is crucial because the factors provide immediate insight into the zeros of the function and their corresponding multiplicities. This form facilitates the process of sketching the graph by highlighting where the function intersects or touches the x-axis and helps in identifying key features of the graph such as turning points and behavior near the zeros.

Zeros and Multiplicity

The zeros of the polynomial are the values of x that make the function equal to zero. The concept of multiplicity indicates how many times a particular zero appears as a factor in the polynomial. Zeros with odd multiplicities typically result in the graph crossing the x-axis, while zeros with even multiplicities lead to the graph merely touching the x-axis and turning back, affecting the shape of the graph near these points.

Intercepts

Intercepts are points where the graph of the function crosses the x-axis and the y-axis. Identifying the x-intercepts is accomplished by setting the polynomial equal to zero, which is simplified in its factored form. The y-intercept can be found by evaluating the function at x = 0, providing a starting point for plotting the graph.

End Behavior Analysis

End behavior analysis examines how the values of the polynomial function behave as x approaches positive or negative infinity. This is largely determined by the leading term in the polynomial, which can be deduced from the product of the factors and their degrees. The sign and degree of the leading term indicate whether the graph rises or falls on either end, an essential aspect of accurately sketching the complete graph.

Transcript

00:01 So remember that when we are trying to sketch a graph of a polynomial, well, the first thing we have to do is find the x intercepts as well as their multiplicity.

00:13 So our first x intercept here is x is equal to negative 1, and this has a multiplicity of 2.

00:20 And so our next one is x is equal to 1, and this has a multiplicity of 3.

00:27 And our last one is x is equal to 2, and this has a multiplicity of 1.

00:32 So remember that an even multiplicity means that we will only touch the x -axis and an odd multiplicity means that we cross the x -axis.

00:43 So now we can look at our end behavior here.

00:45 So when x approaches negative infinity, we have a negative times a negative infinity squared times a negative infinity cubed times a negative infinity.

01:01 And so what this means is that we have we have a negative times a point.

01:05 Positive times a negative times a negative.

01:09 And so that means our polynomial approaches negative infinity because these two negative infinities make a positive, but this negative on the outside makes it negative again.

01:20 So now let's see when x approaches positive infinity.

01:23 Positive infinity.

01:24 So we have this negative on the outside.

01:27 And so we have positive infinity squared.

01:29 So this gives us positive infinity squared...

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