Hi Laverne E
You can review the definition, domain and range of the Secant function in your text or online important points to keep mind is it is a Reciprocal Function that has no x intercepts, as the denominator approaches zero the function approaches negative or positive infinity. Also use Radians
Basically you still enter values for x and calculate values for y per your function. You want to plot a secant with respect to its definition:
secant (x) = 1/cos(x), a reciprocal function as the denominator, the cosine, approaches zero you approach a boundary for the secant. When the cosine is 1 the secant is 1 when the cosine is negative 1 the secant is negative 1.
Of course this means you will have vertical asymptotes in the secant plot where the cosine of x approaches zero.
cos(0) = 1
cos(∏) = -1
But the cos(∏/2) = 0
So as your calculation of (∏4x), based on the x values you enter, approach (∏/2), (3∏/2), (5∏/2) your graph approaches positive or negative infinity
Note below inside f(x) = -4 sec (∏4x)
When x = 0, 1/12, 1/6, 1/4, or 1/2
Then (∏4x) = 0, ∏/3, 2∏/3, ∏ and 2∏
I rearranged so you can see how it multiplies easier and you have
f(0) = -4sec(∏*(4*0) = - 4sec(0)= -4(1/cos(0)) = -4(1/1) = -4
f(1/12) = -4sec(∏*4*(1/12)) = -4sec(∏/3) = -4(1/cos(∏/3)) = -4(1/0.5) = -4(2) = -8
f(1/6) = -4sec(∏*4*(1/6)) = -4sec(2∏/3) = -4(1/cos(2∏/3)) = -4(1/(-0.5)) = -4(-2) = 8
f(1/4) = -4sec(∏*4*(1/4)) = -4sec(∏) = -4(1/cos(∏)) = -4(1/-1) = 4
f(1/2) = -4sec(∏*4*(1/2)) = -4sec(2∏) = -4(1/cos(2∏) = -4(1/1) = -4
But at
f(1/8) = -4sec(∏*4*(1/8)) = -4sec(∏/2) = -4(1/cos(∏/2)) = -4(1/0) this is undefined, a boundary value for your function or a vertical asymptote. Odd multiples of (1/8) are asymptotes for you, 4 and -4 are horizontal boundaries as well.
You can make a table with four columns labeled x values, (4*∏*x) the calculation which give multiples of ∏ and for range or y values -4*(1/cos(4*∏*x) is the calculation and finally the value for y. Remember to substitute the value you calculate for (4*∏*x) in (1/cos(4*∏*x))
You can plot your function at Desmos.com, it will be in Radians unless you change it. To see a comparison you can also plot y = -4cos(∏4x) on the same grid.
A comparison of f(x) = -sec(∏x) and y = —cos(∏x) might be easier to look at and see that whenever the cosine intercepts the x axis the secant approaches positive or negative infinity.
I hope this helps
