Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhib

Sketch the graph of the polynomial function. Make sure your...

Key Concepts

Leading Coefficient and Degree

The degree of the polynomial, which is the highest power of the variable, alongside the leading coefficient, determines the dominant behavior of the function for large absolute values of x. The combined effect informs how the function rises or falls at the ends of the graph.

Multiplicity of Zeros

The multiplicity of a zero refers to how many times a particular zero (or factor) appears in the polynomial's factorization. A zero with an odd multiplicity will have the graph cross the x-axis, whereas a zero with an even multiplicity will cause the graph to touch the x-axis and turn around.

End Behavior

End behavior describes how the values of a polynomial function behave as x approaches positive or negative infinity. This behavior is dictated by the degree of the polynomial and the sign of its leading coefficient, providing insight into the overall direction of the graph’s ends.

Intercepts

Intercepts are the points where the graph of a function crosses the axes. The x-intercepts, or zeros, are found by setting the function equal to zero, and the y-intercept is typically found by evaluating the function at x = 0. Identifying these points helps in accurately sketching the curve.

Polynomial Functions

A polynomial function is a mathematical expression consisting of variables and coefficients that involves only non-negative integer powers of the variable. These functions are continuous and smooth, which means they can be graphed without breaks or sharp corners.

Transcript

00:03 All right, we want to look at the graph of p of x equals 1 4th times x plus 1 cubed times x minus 3.

00:20 So we're going to discuss n behavior and x and y intercepts to get an idea of what's going on with this curve.

00:28 So starting with n behavior, we can determine what's going on with this curve based on the degree of the polynomial.

00:43 So if i were to multiply this all out and expand it, then i would end up with x cubed times x.

00:53 The lead term would be, well, i'd have x cubed times x times one -fourth.

01:00 The lead term would be one -fourth x to the fourth power.

01:08 So that's a polynomial of degree four.

01:11 It's a fourth -degree polynomial.

01:13 The lead coefficient is one -fourth, which is positive.

01:17 So that tells me that as x is approaching negative infinity, as x gets really small, y will be approaching positive infinity, it'll be increasing.

01:27 And as x approaches positive infinity, in other words, as x is moving towards the right, y will also be increasing.

01:35 In other words, the end behavior of this graph is increasing to the right, it's doing whatever it's doing in the middle, and it's increasing to the left and to the right.

01:46 In other words, it's increasing in both directions.

01:48 Both ends are going the same way.

01:51 So the next thing we will talk about then will be our x intercepts, which will occur where the polynomial is equal to zero.

02:05 So i'm going to have one -fourth times x plus one cubed times x minus three equals zero.

02:17 Now we let each factor equal zero.

02:19 You'll have x plus one to the third power equals zero.

02:25 X minus 3 equal to 0, add 3 to both sides, you will get x equals 3.

02:32 If you take the cube root of both sides or factor this, whichever you have x plus 1 times x plus 1 times x plus 1.

02:38 In other words, x plus 1 is equal to 0, which means x must be negative 1.

02:45 Now, what's important about these here to point out is that this root at x equals negative 1 is a triple root, meaning it has.

03:02 Has a multiplicity of three...

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