Find the zeros of each function. Then graph the function....

Key Concepts

Zeros of a Function

The zeros (or roots) of a function are the x-values for which the function evaluates to zero. They represent the points where the graph of the function intersects the x-axis. Finding these involves solving the equation f(x)=0, which, in the case of factorable polynomials, can be done by setting each factor equal to zero.

Polynomial Factorization

Polynomial factorization is the process of expressing a polynomial as a product of its simpler polynomials. This form makes it easier to find the zeros because it allows one to set each factor to zero, thereby solving for the roots. Factoring is a fundamental technique in algebra that simplifies the analysis of polynomial equations.

Cubic Function

A cubic function is a type of polynomial where the highest power of the variable is three. Cubic functions can have up to three real zeros and their graphs typically display characteristic features such as inflection points and turning points. Understanding the general shape and behavior of cubic functions is essential for accurate graphing.

Graphing Polynomial Functions

Graphing polynomial functions involves plotting important features such as zeros, y-intercepts, and local extrema. It requires understanding the overall behavior of the polynomial, including its end behavior and turning points, and using techniques such as factorization to determine key points for an accurate sketch of the graph.

Transcript

00:01 So we've been asked to find the zeros of this function and then use that to help us graph.

00:08 So the first thing we need to do is remember what zeros are.

00:13 So zeros are just the numbers we plug in for x that will give us zero for y.

00:19 So in this function, when we plug something in for x, we're going to get some number here times some number here times some number here.

00:31 We're just going to be multiplying three numbers together.

00:34 I don't know what those numbers are.

00:36 It depends on what we choose for x.

00:38 So for now, i'm just going to call them a, b, and c.

00:43 So i know that a times b times c, i want that to equal zero.

00:53 So i need to figure out what x is i would have to choose.

00:57 To make this equal zero.

01:00 Now that sounds impossible, but there's something special about zero.

01:04 The only way to get zero when we're multiplying is if one of these numbers is zero.

01:12 So one way to get it is to make a zero.

01:15 If a was zero, this would work, b and c can be whatever they want to be.

01:19 Or i would have to make b zero.

01:22 Or i would have to make c zero.

01:24 Or i can make a, b and zero and c not zero.

01:28 I can make all of them zero.

01:30 As long as at least one of them is zero, i will get zero afterwards.

01:34 And there is no other way to do it.

01:36 There aren't any other numbers you can multiply by to get zero.

01:40 One of them has to be zero.

01:44 So either my first number, a was zero.

01:48 If that's true, that means i have to figure out what i could plug in for x to make this whole thing zero.

01:55 So x plus 1, what would x have to be to make this true? well, i would just have to subtract one from both sides, then i get x equals negative 1.

02:10 Same thing.

02:11 What if i wanted this one to be 0? what would make that true? well, if x minus 2 is 0, that means x would have to be 2.

02:25 You might be getting the pattern now.

02:27 Now if i wanted x minus 3 to be 0, well, i would just add 3 to both summits, and then i would find that x would have to be 3.

02:39 So these are the three numbers that will help us, that will give us zeros.

02:48 So now we're going to use that to help us graph.

02:55 The fact that these are zeros tells us that this is negative 0, because we know the whole point of it being 0 was that it makes the y -4ordinate 0.

03:04 This is 2 -0, and this is 3 -0...

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