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

step 1 All right. We're going to sketch a graph for the function. P of X equals x minus 1 squared times

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

Key Concepts

Graph Sketching Strategies

Sketching the graph of a polynomial involves combining knowledge of zeros, intercepts, multiplicity, end behavior, and potential turning points. This systematic approach ensures that key characteristics of the polynomial are accurately represented, providing a clear and complete depiction of its graph based on the function's algebraic properties.

End Behavior

The end behavior of a polynomial function describes how the graph behaves as x approaches positive or negative infinity. This behavior is determined by the degree of the polynomial and the sign of its leading coefficient. For instance, a polynomial of degree 3 (cubic) will have different behavior at each end of the graph, which is critical when sketching the overall shape of the function.

Zeros and Intercepts

The zeros of a polynomial are the solutions to the equation P(x) = 0 and are directly related to the x-intercepts of its graph. In a factored polynomial, these zeros appear as factors set to zero. Identifying the zeros provides the key points where the graph intersects the x-axis, and calculating the y-intercept (by evaluating the polynomial at x = 0) gives another crucial point through which the graph passes.

Multiplicity of Zeros

The multiplicity of a zero in a polynomial function describes how many times that zero appears as a factor. It affects the graph's behavior at the intercept: zeros with an odd multiplicity cross the x-axis, while those with an even multiplicity cause the graph to touch and turn around at that point. Understanding multiplicity is essential for accurately sketching the graph near its intercepts.

Transcript

00:03 All right.

00:04 We're going to sketch a graph for the function.

00:15 P of x equals x minus 1 squared times x minus 3.

00:25 So the three main things that we want to consider are n behavior, x intercepts, and y intercepts.

00:40 That's the main information we're going to find, i think, to get an idea of what's going on here.

00:50 So let's think about n behavior first.

00:54 This is kind of factored out for us right now.

00:57 But if we were to multiply this all out, i have x minus 1 squared.

01:01 So this would be an x squared times x.

01:07 So we would end up with a lead term of x to the third power.

01:15 So we have a polynomial whose degree is 3 and whose lead coefficient is going to be positive.

01:26 So that tells us that the end of our graph, as x is approaching negative infinity, y would be approaching negative infinity.

01:41 In other words, as x is decreasing, y is decreasing.

01:44 And as x gets very large, as x approaches positive infinity, y will also be getting very large.

01:51 Why is approaching positive infinity? so as x is increasing, y is increasing.

01:55 In other words, we have a graph.

01:59 That is x goes to the left, y will be decreasing.

02:03 Whatever it's doing in the middle, it's doing, and then as x gets very large, y has to be increasing.

02:13 So the next thing we're going to find then is our x intercepts.

02:21 Now, this was nicely factored for us already.

02:25 What we want to do is take this function and let it equal zero.

02:30 So x minus 1 squared times x minus 3 squared would equal zero.

02:37 So we let, oh, excuse me, i just, i lost check what i was doing for a second.

02:43 That is not x minus three squared.

02:45 X minus one squared times x minus three equals zero.

02:48 So now we want to let each factor equal zero.

02:51 We will have x minus one squared equal zero and x minus three equals zero.

03:00 If i add three to both sides, i get x equals three.

03:07 And if i took the square of each side here, x minus one would be zero.

03:12 So you get x equals 1.

03:14 Now, these roots tell us quite a bit also.

03:21 This right here, this root at x equals 1 is what we call the double root...

Please give Ace some feedback