Question

f is a trigonometric function of the form f(x) = a cos(bx+c)+ d.
Below is the graph of f(a). The function has a maximum point at (-4, 2) and a minimum point at (-1,-3).
Find a formula for f(x). Give an exact expression.

Answer 1

$f(x) = a \cos(bx + c) + d$

Where a is the amplitude

b is used to find the period

$\frac{2\pi}{b}$

The phase shift can be fined by doing

$bx + c = 0$

then the midline is y=d.

Finding A: The amplitude is the half of the distance between the. y values of the max and the min.

$\frac{2 - ( - 3)}{2} = 2.5$

So a=2.5

Note: A is Positve never negative so if you get a negative a, take the absolute value.

Period:

Find the distance between x values,

$- 1 - ( - 4) = 3$

The distance between the x values of the max and min x values is half the distance of the period. So the period is 6.

$\frac{2\pi}{b} = 6$

$b = \frac{\pi}{3}$

Note : If b is negative, take the absolute value.

Finding the midline.

To find the midline find the midpoint of the max and min y values.

$\frac{2 + ( - 3)}{2}$

$\frac{ - 1}{2}$

So our midline is

$- 0.5$

So as of right now, our trig formula is

$2.5 \cos( \frac{\pi}{3}x ) - 0.5$

Plug in any x or y value,

$2.5 \cos( \frac{\pi}{3} ( - 4)) - 0.5 = 2$

$2.5 \cos( \frac{ - 4\pi}{3} ) = 2.5$

$\cos( \frac{ - 4\pi}{3} ) = 1$

This is not right so we need to shift this to the nearest value that is right

If we subtract,

$\frac{ - 2\pi}{3}$

We will have a cosine value of 1.

So our phase shift is

$2.5\cos( \frac{\pi}{3} x - \frac{2\pi}{3} ) - 0.5$

Our phase shift would be 2 units to the right.

$2.5 \cos( \frac{\pi}{3} (x - 2)) - 0.5$