Question

In what direction and by how many units is the graph of f(x) = 6 sin(2x + π) − 5 vertically and horizontally shifted? (1 point)

Group of answer choices

Down 5, left pi over 2

Up 5, left pi over 2

Up 5, right pi over 2

Down 5, right pi over 2

Answer 1

The first option:

Down 5, left pi/2.

Explanation:

Let's define the translations in a general way.

Horizontal shift.

For a function f(x), a horizontal shift of N units is written as:

g(x) = f(x + N)

if N is positive, the shift is to the left.

if N is negative, the shift is to the right.

Vertical shift.

For a function f(x), a horizontal shift of N units is written as:

g(x) = f(x) + N

If N is positive, the shift is upwards.

If N is negative, the shift is downwards.

In this case, we have the function:

f(x) = 6*Sin(2*x + π) - 5

Where the original function was something like:

h(x) = 6*sin(2*x)

Here we can already see the vertical shift is of minus 5 units, so the shift is 5 units down.

then we get:

f(x) = h(x) - 5 = 6*sin(2*x) - 5

Now we can apply a shift to the left of π/2 units, so we get:

f(x) = h( x + π/2) - 5 = 6*sin(2*(x +π/2) ) - 5

f(x) = 6*sin(2*x +π ) - 5

Then the correct option is:

Down 5, left pi over 2