Match the polynomial function with one of the graphs I-VI. Give reasons for your choice. U(x)=-x

Match the polynomial function with one of the graphs I-VI....

Key Concepts

Polynomial Functions

A polynomial function is an expression consisting of variables and coefficients that involves only non-negative integer powers of the variable. The degree of the polynomial and the sign of its leading coefficient determine many of its basic properties, such as its shape and end behavior.

Zeros and Roots

Identifying the zeros or roots of a polynomial involves solving the equation where the function equals zero. These zeros indicate where the graph intersects the x-axis, and their multiplicities reveal whether the graph merely touches or actually crosses the axis at those points.

End Behavior Analysis

The end behavior of a polynomial is determined by its leading term, which indicates how the function behaves as the input values become very large or very small. This concept helps in predicting whether the graph will rise or fall as x approaches infinity or negative infinity.

Critical Points and Turning Points

Finding critical points by analyzing the derivative of the polynomial is essential for determining local extrema. These turning points highlight where the graph changes direction and are crucial for matching the curvature and peaks or valleys in a given graph.

Graphical Analysis

Graphical analysis involves synthesizing information about the polynomial’s zeros, end behavior, and turning points to determine which graph best represents the function. This process emphasizes the importance of comparing a function’s theoretical properties with visual representations.

Transcript

00:04 For this question, we want to explore the graph of view of x, which is equal to negative x cubed plus 2x squared.

00:18 So a couple of things that we need to consider to identify the graph of this function.

00:25 First, we will discuss n behavior, which tells us what the graph is doing as x gets very small and very large.

00:37 And what we want to notice is that the degree is three.

00:42 It's a third degree polynomial.

00:44 It's the highest exponent that we have here.

00:47 So it's an odd degree polynomial.

00:50 That indicates that our ends are going into opposite directions.

00:53 And because the lead coefficient is negative, that means that as x is going to the left towards negative infinity, y will be approaching positive infinity, or in other words, y is going to be increasing.

01:11 And as x goes to the right, as x approaches positive infinity, y will be approaching negative infinity.

01:19 In other words, i've got a graph that's going up to the left, whatever it's doing in the middle, and it's going down to the right.

01:28 That's ultimately what this graph is doing, again, as x gets very large or as x kids very small.

01:34 So the next thing we want then is to talk about our intercepts, x and y intercepts, x and y intercepts, sometimes called the roots, or the zeros are going to be where the function is equal to zero.

01:48 So if we let negative x cubed plus 2x squared equal zero, i can now factor out a greatest common factor of negative x squared.

02:03 So negative x cubed divided by negative x squared is x.

02:06 2x squared divided by negative x squared will be negative 2.

02:10 So now we let each factor equals 0...

Please give Ace some feedback