Question

A sine function is transformed such that it has a single x-intercept in the interval (0,pi), a period of pi and a y-intercept of 3
a) determine an equation for a function that satisfies the properties given.

Much help is needed !! I only got k=2 but I’m not sure how to find the rest of the variables.

Answer 1

Just to make sure we're using the same language, I'm going to use the function form of:

$y = A\sin(kx) + h$

[] I would agree that k = 2, since the period is only half as long as a normal sine function. So, we so far, have y = A sin(2x) + h. We still need to find A and h.

[] The y-intercept is 3. Remember that the y-intercept happens when x = 0. So, plugging in x = 0 into our formula, we have: 3 = A sin(2*0) + h. In other words, 3 = A sin(0) + h = 0 + h = h. So, we now know that h = 3. The formula is now y = A sin(2x) + 3.

[] Finally, there is a single x-intercept. Picture what the sine function looks like right now, it is floating in the air around y = 3. We need to stretch it vertically until it just grazes the x-axis. 

If A = 1, then our sine function bounces between 2 and 4 (+/- 1 around h = 3). But that doesn't touch 0, so no good.

If A = 2, then our sine function bounces between 1 and 5 (+/- 2 around h = 3). Again, not quite touching 0 yet.

The answer should be A = 3, then our sine function bounces between 0 and 6 (+/- 3 around h = 3).

The final formula is y = 3 sin(2x) + 3.