What are the key features of these polynomial functions?

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In summary, the first function has a y-intercept of 8 and an x-intercept of -2. The second function has a y-intercept of 30 and an x-intercept of 1, 2, 3, 4.
  • #1
first function: f(x)= x^3-3x^2-6x+8
2nd function: f(x)= x^4-10x^3+35x^2-50x+24

1)whats y intercepts
2)whats x intercepts (zeros)
3)turning points
4)table and graph with appropriate scales
5)intervals of increase/decrease after you determined turning points
6)end behavior
7)symmetry: odd or even

for numbers 1 and 2 i think you have to put the functions onto the graphing calculator and find the y and x intercepts. i did that and i got 8 for the y intercept, -2, 1, 4 for the x intercept of the first function. for the 2nd function i think the y intercept is 30 and the x intercept is 1, 2 , 3, 4. are those intercepts right? that is by far all i know about this whole problem. i know how to graph the functions but how do i make table? I need help on turning points, intervals, end behavior, and symmetry of both functions. thanks for helping.
 
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  • #2
To find the y-intercept plug-in x = 0. (Your answer is incorrect for the second function.)

Do you know what a turning point is? Your graphing calculator should be able to help you find these.

Do you know what it means for a function to be increasing? ... decreasing?
 
  • #3
mathandphysic said:
first function: f(x)= x^3-3x^2-6x+8
2nd function: f(x)= x^4-10x^3+35x^2-50x+24

1)whats y intercepts
2)whats x intercepts (zeros)
3)turning points
4)table and graph with appropriate scales
5)intervals of increase/decrease after you determined turning points
6)end behavior
7)symmetry: odd or even
For #6, look up Leading Term Test for Polynomials.
For #7, are you supposed to find the symmetry by looking at the graph, or algebraically? If algebraically, this involves finding f(-x).
 
  • #4
One problem is that you have put these into the 'precaculus' section. While, if you are able to completely factor the polynomials, you can "reason out" the answers, it is far simpler to find first and second derivatives to answer the question. Was this question specifically for a "PreCalculus" class or do have you had "derivatives"?
 

1. What is the purpose of analyzing two functions?

The purpose of analyzing two functions is to compare and contrast their behaviors and characteristics, such as their domain, range, and rate of change. This can help in understanding how the functions are related and how they may be used in real-world situations.

2. How do you graph two functions on the same coordinate plane?

To graph two functions on the same coordinate plane, plot the points for each function separately and then connect them with a line. Make sure to label each function and use different colors or styles for each line to differentiate them.

3. What is the difference between a linear and a quadratic function?

A linear function is a straight line with a constant rate of change, while a quadratic function is a curve with a changing rate of change. Linear functions have a constant slope, while quadratic functions have a changing slope.

4. How do you determine if two functions are inverses of each other?

To determine if two functions are inverses, you can use the horizontal line test. If a horizontal line intersects both graphs at the same point, then the functions are inverses of each other. Another way is to find the composition of the two functions and see if it results in the input value.

5. Can two functions have the same graph?

Yes, two functions can have the same graph. This is known as a one-to-one function, where each input has a unique output. However, it is important to note that two functions can have the same graph but have different domains. In this case, they are not considered to be the same function.

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