Graphing Polynomial Functions: Finding x-Intercepts

In summary, the rule for determining when a graph crosses the x-axis at an x-intercept or when it just touches and turns away from the x-axis is to look at the powers of the factors. If the power is even, the graph will intersect the x-axis, and if the power is odd, the graph will only touch and turn away from the x-axis. This can be seen by graphing each function and observing the behavior at the x-intercepts.
  • #1
Buddah
7
0
Graph each function given below on a graphing calculator to find a general rule for determining when a graph crosses the x-axis at an x intercept or when the graph just touches and turns away from the x axis. State the rule that you find.

y = (x + 1)^2(x - 2)

y = (x - 4)^3(x - 1)^2

y = (x - 3)^2(x + 4)^4
 
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  • #2
1) This is a homework problem, so it should belong in the homework forums
2) You should post an attempt of the problem
3) What do you expect from us really? That we graph it for you? :confused: We can't graph anything on this forum (yet)...
 
  • #3
Agreed with micromass, but here's a hint. Look at the numbers carefully.
 
  • #4
For graphing you can use "Microsoft Mathematics". Its really a awesome software.
Give it a try. :smile:
 
  • #5
okay well i tried to do it

is this correct?

x intercept y=0

y = (x + 1)^2(x - 2)
0 = (x + 1)^2(x - 2)
(x + 1)^2 = 0 =====> x + 1 = 0 =====> x = -1 (-1 , 0)
x - 2 = 0 =====> x = 2 (2 , 0)

y = (x - 4)^3(x - 1)^2
0 = (x - 4)^3(x - 1)^2
(x - 4)^3 = 0 ====> x - 4 = 0 =====> x = 4 (4 , 0)
(x - 1)^2 = 0 ====> x - 1 = 0 ====> x = 1 (1 , 0)

y = (x - 3)^2(x + 4)^4
0 = (x - 3)^2(x + 4)^4
(x - 3)^2 = 0 ====> x - 3 = 0 ====> x = 3 (3 , 0)
(x + 4)^4 = 0 ===> x + 4 = 0 ====> x = -4 (-4 , 0)
 
  • #6
Am having trouble explaining it
 
  • #7
From what I understand of your original post, you do not need to do all that work. It's simple, look at the graphs.

Hint: Take notice of the powers (not just by degree:wink:) to determine whether it goes right through or only touches.

EDIT: By the way, recall what multiplicity is.


The last function might be a little hard to view, but that should not affect your rule.:smile:

It might help to recognize how x2 and x3 look like.
 
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  • #8
Yes it's correct, now try plugging in some numbers that make the factors negative :)
 

FAQ: Graphing Polynomial Functions: Finding x-Intercepts

What is a polynomial function?

A polynomial function is a mathematical function that can be written in the form f(x) = anxn + an-1xn-1 + ... + a1x + a0, where an through a0 are constants and x is the variable. It is a type of algebraic function that involves powers and coefficients of a variable.

How do I graph a polynomial function?

To graph a polynomial function, you can plot several points on the coordinate plane by substituting different values for x in the function and solving for y. You can also use the leading coefficient, degree, and end behavior of the function to determine the general shape of the graph. Once you have plotted enough points, you can connect them with a smooth curve to create the graph of the polynomial function.

What are x-intercepts?

An x-intercept is a point on the graph of a function where the graph crosses or touches the x-axis. It is the value of x at which the function has a y-value of 0. In other words, it is a value of x that makes the function equal to 0. On a graph, x-intercepts are represented by points on the x-axis.

How can I find the x-intercepts of a polynomial function?

To find the x-intercepts of a polynomial function, you can set the function equal to 0 and solve for x. This will give you the x-values at which the function crosses the x-axis. You can also use a graphing calculator or software to find the x-intercepts by graphing the function and looking for the points where the graph crosses the x-axis.

Why are x-intercepts important for polynomial functions?

X-intercepts are important for polynomial functions because they give us information about the roots or solutions of the function. They can help us determine the number of real solutions to an equation, the behavior of the graph, and the domain and range of the function. Additionally, x-intercepts can be used to find the factors of a polynomial function, which is useful for solving equations or simplifying expressions.

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