Which function is represented in this graph?
2 answers:
Answer:
y = tan (1/2 x - π/2) ⇒ the answer is (b)
Step-by-step explanation:
* Lets talk about the tangent function
- The tangent has a period of π, with each period separated by
a vertical asymptote. The concept of "amplitude" doesn't
really exist because -∞ < y < ∞
- You can transform the graph for tangent
∵ y = A tan B(x + C) + D
* Where:
# A, changes amplitude. (A always = 1)
# B determines the period.
# C determines how the graph is shifted from left to right.
(horizontally)
# D determines how the graph is shifted up or down.(vertically)
∵ The basic function is y = tanФ has an amplitude of 1.
It has a period of π. It has no phase or vertical shifts,
because it is centered on the origin.
* Look to the problem
- From the graph:
- The period is 2π
∵ Period = π/B
∴ 2π = π/B ⇒ ÷ π
∴ 2 = 1/B
∴ B = 1/2
- the starting point of the function is at π instead of zero
∴ It moves π units to the right
∴ C = -π
∴ y = tan 1/2(x - π)
∴ y = tan (1/2 x - π/2)
* The function is ⇒ y = tan (1/2 x - π/2)
* The blue represents y = tanx
The red represents y = tan (1/2 x - π/2)
You might be interested in
Answer:
35 years
Step-by-step explanation:
It will make this problem a whole lot easier if we call the year 1990 our initial year, or year 0; that means that the year 2000 is year 10. The coordinates then that result from that change are (0, 471.56) and (10, 492.53). Population growth is always exponential, and the formula for that, in terms of x and y, where x is the year and y is the population in millions, is
a is the initial population at year 0 and b is the growth rate. If the growth rate is a decimal (less than 1) what we have is decay as opposed to growth. In order to determine when (x) the population (y) is 492.53, we have to find the model by solving for a and b using the coordinates. Once we find this model, we will use it to solve for x by filling in the y of 550 million. So here we go.
Using the first coordinate, we know that at year 0 the population is 471.56. So that is our initial population, a in our formula. Using that value of a and plugging it in to the formula with the other coordinate gives us
Now we have to solve for b. Begin by dividing both sides by 471.56 to get
"Undo" that 10th power by taking the 10th root of both sides which gives you that b = 1.004360381
Now we have our model:
We will plug in a y value of 550 and solve for x using natural logs.
Begin by dividing both sides by 471.56 to get
In order to solve for x we have to pull it down from the position in which it is currently sitting, which is an exponent. Taking the natural log of both sides will solve that problem for us:
The power rule for natural logs tells us we can pull the exponent down in front giving us:
Now we will divide both sides by ln(1.004360381) to isolate the x. Doing that and at the same time finding those values on our calculator gives us:
so
x = 35