Determine the equation of the graph in the form y = acos(bx).

I will try and describe the Graph.
-It starts at the origin (0,0)
-It has a minimum value of -2
-It has a maximum value of 2.
-It has an x-intercept of 90 degrees.
-It starts moving downward below the x-axis, and then curves back upwars, and continues in this pattern.

My answer:
I think "a" should= 2.5, but I do not not what "b" should be. Whatever I put in, I can not get the graph to start at the origin. It always starts above or below y=0.

Could someone please help me?
Thanks.

A cosine curve without a phase shift cannot start at (0,0)

The curve that would fit your description would have to be
y = 2cos 2(x + 45 degrees)

Check:
(0,0) lies on it.
when x = 90, y = 2 cos 270 = 0
so there is an x-intercept at 90
when x=45, y = 2 cos 180 = -2
when x = 135, y = 2 cos 360 = +2

the period of the curve is 180 , shown by b=2

To determine the equation of the graph in the form y = acos(bx), we can use the given information. Let's break down the problem step by step.

Step 1: Determine the amplitude (a)
The maximum value of the graph is 2, so the amplitude (a) should be half of that value. Therefore, a = 2/2 = 1.

Step 2: Determine the period (T) or find the value of b
The x-intercept occurs at 90 degrees, which corresponds to a quarter of the period. To find the whole period, we need to multiply it by 4. Therefore, the period (T) is equal to 4 * 90 degrees = 360 degrees.

The formula to find the value of b is b = 2π/T, where T is the period in radians.

Converting 360 degrees to radians:
360 degrees = (360 * π) / 180 = 2π radians

Now, let's substitute the value of T into the formula:

b = 2π / 2π
b = 1

Therefore, b = 1.

Step 3: Write the equation
Now that we have the values of a and b, we can write the equation in the form y = acos(bx):

y = 1 * cos(1x)

Final equation: y = cos(x)

So, the equation of the graph in the form y = acos(bx) is y = cos(x).

To determine the equation of the graph in the form y = acos(bx), we can use the given information about the graph. Let's analyze the given details:

1. It starts at the origin (0,0): This means that the value of the y-intercept is 0, which implies that cos(0) = 1. Therefore, the equation becomes y = a * cos(bx) + 0, which simplifies to y = a * cos(bx).

2. It has a minimum value of -2: Since the minimum value of cos(x) is -1, we can set a = -2. This gives us y = -2 * cos(bx).

3. It has a maximum value of 2: The maximum value of cos(x) is 1, so we need to incorporate a scaling factor of 2. This gives us y = 2 * cos(bx).

4. It has an x-intercept of 90 degrees: The x-intercept occurs when y = 0. Substituting this into the equation, we get 0 = 2 * cos(b * 90). Solving for b, we have cos(90b) = 0. This means that 90b is an odd multiple of 90 degrees, so b = (2n + 1) / 2, where n is an integer.

To summarize, the equation of the graph is y = 2 * cos[((2n + 1) / 2) * x].