Question

Just the answer please.
1: http://imgur.com/OV8dUWz

2: http://imgur.com/vnB3uVZ

Answer 1

Answer on Question #58626 – Math – Trigonometry

Question

Which function’s graph has a period of 2?

$$\begin{array}{l} y = 2 \sin \pi x \\ y = 3 \cos x \\ y = -4 \sin 2x \\ y = \cos\left(x - \frac{\pi}{2}\right) \\ \end{array}$$

Solution

Let the unknown sine function be $y = A \sin(bx + c) + d$

Then the period: $T = \frac{t}{b}$, where $t$ is a regular period of function (for example, regular period of sine and cosine $t = 2\pi$ and for tangent and cotangent $t = \pi$).

Our job is to find a function whose period is equal to 2. So in our formula, $T = 2$. From the question it is clear that we need to look for these functions among the sine and cosine, then $t = 2\pi$.

Substitute the values into the formula:

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

hence

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

We also know that $b$ is the coefficient of $x$, so we now find the function whose coefficient near $x$ is $\pi$. That’s only one function:

$$y = 2 \sin \pi x$$

**Answer:**

$$y = 2 \sin \pi x$$

Question

Which description matches the transformation $y = \cos x$ undergoes to produce $y = 3\cos (-2x)$ ?

Reflection through the y-axis, vertical shift of 2 units, horizontal shift right by 3 units.

Horizontal shift left 2 units, then vertical shift up by 3 units.

Horizontal compression by factor $\frac{1}{2}$ , vertical stretch by factor 3, then a reflection through the y-axis.

Horizontal stretch by factor 2, reflection through the x-axis, then vertical stretch by factor 3.

Solution

Transformations "after" the original function.

Transformations "before" the original function.

Suppose that we have a function $y = \cos x$ . To transform it, we do the following steps:

1. $y = \cos (2x)$ : we have to shrink the function $y = \cos x$ horizontally by $\frac{1}{2}$ .

2. $y = 3\cos (2x)$ : the function $y = \cos (2x)$ is stretched vertically by 3.

3. $y = 3\cos (-2x)$ : the final step is to flip the function $y = 3\cos (2x)$ over the y-axis.

Answer:

Horizontal compression by factor $\frac{1}{2}$ , vertical stretch by factor 3, then a reflection through the y-axis.

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