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. [3]
y = (x + 1)^2(x - 2)
y = (x - 4)^3(x - 1)^2
y = (x - 3)^2(x + 4)^4
i solved it:
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)
however am having trouble explaining it
you've missed the point of the original question ...
did you graph each function? does the graph of each function cross the x-axis at each zero (x-intercept), or does it just "touch" the x-axis and turn away?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
which zeros cross? which turn away?
That's correct. So because (x+1) is squared. If we look on the graph, it bounces off the x axis, it doesn't touch it.
Now have a look at the second graph:
This graph is
You already worked out that there's x-intercepts at 4 and 1. Do you notice any trends? Since the x-1 is squared is bounces off the graph....
I'm not sure if you've studied cubics before but a perfect cube is one of the most simple examples of a graph that has what we call an "inflection point". So at x=4 there is an inflection point.
Can you make any conclusions from this?
lol, don't be shy...
Here is a standard cubic graph (you don't have to know this but I'll show you anyway...):
As you can see the "straight" part in the middle is what we call the "Point of Inflection". This is all you need to know, that when there is a shape like that it is called a point of inflection.
Now if you compile all the stuff you've learnt from the 2 graphs I have shown you in my previous 2 posts can you come to any conclusions?
I'll start you off...
If a factor is (x-1) it will cut the x axis
If a factor is it will....
If a factor is it will...
If you can't then you should probably get your teacher to help you because it will make it easier if you have some one on one time with a real teacher