Question

Select the graph of y = 3 sin (x - 1).

Answer 1

Answer:

D.

Step-by-step explanation:

The graph of 3 sin x will stretch vertically sin x by a factor 3. So 3 sin x will have a maximum value of 3 and a minimum of -3.

So it is either C or D.

The  -1 in the parentheses will move the graph 1 unit to the right and as the graph of y = sin x passes through the origin, the required graph will pass through (1, 0) - so it  is Graph D.

Answer 2

Answer:

D.

Step-by-step explanation:

We can cancel out choice A and choice B because both graphs' amplitude are 2. (Amplitude determines the max-min so if amplitude is 2 then max is 2 and min is -2)

Now for graph C and D, if we do not know how does sine graph shift, we will do substitution to find value of y when x = a.

Let's try x = 0 because both graphs have different y-value for x = 0.

$\displaystyle \large{y = 3 \sin(x - \frac{\pi}{4} ) } \\ \displaystyle \large{y = 3 \sin(0 - \frac{\pi}{4} ) } \\ \displaystyle \large{y = 3 \sin( - \frac{\pi}{4} ) }$

Let's recall the negative measure;-

$\displaystyle \large{ \sin( - x) = - \sin(x) }$

Therefore:-

$\displaystyle \large{y = - 3 \sin( \frac{\pi}{4} ) } \\ \displaystyle \large{y = - 3 \cdot \sin( \frac{\pi}{4} ) }$

We know that π/4 is equivalent to 45° because π is defined as 180° and 180°/4 is 45°

Using hand method, sin(45°) is √2/2

$\ \displaystyle \large{y = - 3 \cdot \frac{ \sqrt{2} }{2} } \\ \ \displaystyle \large{y = \frac{ - 3\sqrt{2} }{2} }$

Looks like when we substitute x = 0, the y-value becomes negative.

The only graph that's reasonable is D.

Another method is you substitute x = π/4 in which would make y-value = 0.

Since π/4 is between 0 < x < π/2, the graph D is correct because graph C may have amplitude of 3 but y-value does not become 0 when substitute x = π/4

So D.