Question

Help with Polynomial Functions!

Answer 1

Esmeralda and Heinz both work in a science lab. In order to secure funding for their future experiments, they must present their findings to some investors. The investors are not interested in listening to formulas. They want to see graphs because they are visual people. Unfortunately, Esmeralda and Heinz are having some difficulties.

Esmeralda and Heinz are working to graph a polynomial function, f(x). Esmeralda says that the third-degree polynomial has four intercepts. Heinz argues that the function only crosses the x-axis three times. Is there a way for them both to be correct? Explain your answer.

Heinz has a list of possible functions. Pick one of the g(x) functions below, show how to find the zeros, and then describe to Heinz the other key features of g(x).
g(x) = x3 – x2 – 4x + 4
g(x) = x3 + 2x2 – 9x – 18
g(x) = x3 – 3x2 – 4x + 12
g(x) = x3 + 2x2 – 25x – 50
g(x) = 2x3 + 14x2 – 2x – 14

Provide a rough sketch of g(x). Label or identify the key features on the graph.

Esmeralda is graphing a polynomial function as a parabola. Before she begins graphing it, explain how to find the vertex. Make sure you include how to determine if it will be a maximum or minimum point. Use an example quadratic function to help you explain and provide its graph.

Heinz boasts that he can predict the degree of a polynomial function just by looking at the end behavior. Can Heinz do this? Explain.

Answer 2

Can someone walk me through this problem step by step? I don't want you to tell me what to write though. I need to learn this myself.

Answer 3

"Esmeralda says that the third-degree polynomial has four intercepts."
A third-degree polynomial has at the most three real roots and so it can cross the x-axis at the most three times. So max. x-intercepts is 3.
But here it says four intercepts which could include BOTH x and y intercepts. So both could be correct.

Answer 4

Okay, I think I understand. The Maximum x-intercept is 3 but it could be both x and y intercepts.

Answer 5

correct.

Answer 6

Okay so that fixes the problem I had with the first part I thought it was only the X-intercepts.

Answer 7

I know what a Zero is but I don't know how to find it,

Answer 8

IF Esmeralda said: "the third-degree polynomial has ** four X intercepts **"
then Esmeralda will be wrong and Heinz will be correct.

Answer 9

I never knew how the g(x) and f(x) thing worked.

Answer 10

Are you allowed to use a graphing calculator to solve the third degree polynomial?

Answer 11

Yes.

Answer 12

I don't know how to use them though.

Answer 13

Take the first function \[\Large g(x) = x^3 – x^2 – 4x + 4\]Graph it and find where it crosses the x-axis and those will be the roots or zeros of g(x).
You can use this online graphing website. Type x^3 – x^2 – 4x + 4 on the left column and it will plot it for you.

Answer 14

https://www.desmos.com/calculator

Answer 15

It crosses the x axis at -2, 1, and 2.

Answer 16

Yes those are the zeros: x = -2, x = 1, x = 2

Answer 17

By rough sketch should I just insert my graph?

Answer 18

"I never knew how the g(x) and f(x) thing worked."

f(x) means function of x. The value of f depends on the value of x.
f(x) = x + 2
Here, the left hand side f depends on the value of x.
If x = 1, f(1) = 3. If x =2, f(2) = 4, If x = 5, f(5) = 7.

g(x) is just another function of x.

When plotting the f(x) or g(x) will be along the y-axis and the x will be along the x axis.

Answer 19

Rough sketch implies a hand drawing either with a pencil or an online freehand drawing tool. You can just copy the graph from the online plot that you just did. Make sure to first mark the important points on the graph. The 3 points where it crosses the x-axis and the 1 point where it crosses the y-axis. Then draw a curve through those 4 points similar to the actual graph.

Answer 20

Okay, so I've got that done.

Answer 21

For the next part, pick a sample polynomial that represents a parabola. It should be a second degree polynomial. For example, you can make 2 and 4 its zeros by choosing the polynomial to be (x-2)(x-4). Multiply it out. (x-2)(x-4) = x^2 -6x + 8.
So the function can be: x^2 - 6x + 8.

Answer 22

The general equation of a parabola is" y = ax^2 + bx + c.
If "a" is positive, the parabola will be a vertical parabola that opens upward. Therefore it will have a u-shape and hence will have a minimum at the vertex.
If "a" is negative, the parabola will be a vertical parabola that opens downward. Therefore it will have an inverted u-shape and hence will have a maximum at the vertex.

In the example, x^2 - 6x + 8, "a" is +1 and so it will be a vertical parabola that opens upward and hence will have a minimum at the vertex.

The vertex can be found by completing the square.
Rewrite ax^2 + bx + c as a(x-h)^2 + k by completing the square. (h,k) will be the vertex.

Answer 23

thx @ranga

Answer 24

@ranga , for the second to last question, u said that it should be a second degree polynomial. your answer was a second degree trinomial