Finding the x-intercepts of y=A cos Bx?

Question

I'm working on the question y= -1/3 cos 2pix

I found |A| and the period, but I can't figure out how to find the x-intercepts. I thought I had them, but apparently (0,0) isn't a point on the graph.

I'm not using a graphing calculator and don't expect this to be solved - I'd just like the steps so I can figure it out on my own! :D

Answer 1

Hi,

In the equation y = A cos(Bx), you are correct that |a| is the amplitude. The fact that "A" is negative means the graph actually is reflected or flipped upside down. A normal cosine graph starts on top of a hill at 0°, crosses the x axis down to a valley and then recrosses the x axis back up to the top of a hill in its period. Normally the graph goes to a height of 1 for the hilltop and down to -1 for the valley. With "A" = -⅓, the amplitude is now ⅓ instead of 1, and the graph goes from valley to hill and back to a valley through 1 period.

Now a normal period for a cosine graph is 2π. When there is a "B" number, the new period is 2π divided by the value of "B". For this equation the period is 2π/(2π) = 1. For sine or cosine graphs, I always divide the period by 4 to get the "jump". The "jump" here is 1 ÷ 4 = ¼. The "jump" is the horizontal distance you must move to go from a hill to a zero to a valley to a zero and back to a hill. Because this graph got flipped over it goes from That means instead it goes from a valley to a zero and to a hill to a zero and back to a valley. This graph would start at (0,-⅓). It would move over a jump of ¼ for a zero at (¼,0) and then another jump of ¼ to a hill at
(½, ⅓). Then after the next jump of ¼ , there is a zero at (¾, 0) and finally, after the 4th jump of ¼, there is a point at (1,-⅓).

This point is the end of one period of the graph. the period would repeat from 1 to 2 and 2 to 3, etc.

I hope that helps!! :-)

Answer 2

y = A cos B x
. . . where A is the amplitude or radius
. . . . . when A = 1 the highest point is 1, if A=2, highest . . . . . . . point is 2
. . . . . B is a factor to angle x
the curve intersect the x axis once Bx = 90 or multiple of 90
if B = 1. the curve will intersect the x-axis at x=90, 270, 450 deg
if B = 2. . the curve will intersect the x-axis at x= 90/2 = 45 and
. . . . . multiple of 45, say 90, 135, 180

Answer 3

y=0 for cos Bx=0 so Bx= (2n+1)pi/2 (n=0,1,2......integer)
so
x=(1/B)*(2n+1)pi/2