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How does changing the function from f(x) = −5 cos 2x to g(x) = −5 cos 2x − 3 affect the range of the function?

The function shifts down 3 units, so the range changes from −1 to 1 in f(x) to −4 to −2 in g(x).

The function shifts down 3 units, so the range changes from −5 to 5 in f(x) to −8 to 2 in g(x).

The function shifts down 5 units, so the range changes from −1 to 1 in f(x) to −6 to −4 in g(x).

The function shifts down 5 units, so the range changes from −5 to 5 in f(x) to −10 to 0 in g(x).

Respuesta :

carlosego
For this case we have the following function:
 f (x) = -5 cos 2x
 The range of the function is given by:
 [-5, 5]
 We apply the following transformation:
 Vertical translations:
 Suppose that k> 0
 To graph y = f (x) -k, move the graph of k units down.
 We have then:
 g (x) = -5 cos 2x - 3
 Therefore, the range of the function changes to:
 [-8, 2]
 Answer:
 The function shifts down 3 units, so the range changes from -5 to 5 in f (x) to -8 to 2 in g (x).

Answer:

The correct option is 2.

Step-by-step explanation:

The general form of a cosine function is

[tex]y=A\cos (Bx+C)+D[/tex]

Where, a is amplitude, [tex]\frac{2\pi}{B}[/tex] is period, C is phase shift and D is midline or vertical shift.

The given functions are

[tex]f(x)=-5\cos 2x[/tex]

[tex]g(x)=-5\cos 2x-3[/tex]

In f(x), D=0, it means midline is y=0. In g(x), D=-3, it means midline is y=-3. The graph of function f(x) shifts 3 units down to get the graph of g(x).

We know the the range of cosine function is [-1,1].

[tex]-1\leq \cos 2x\leq 1[/tex]

Multiply both the sides by -5. If we multiply or divide the inequality, then the sides of inequality is changed.

[tex]5\geq -5\cos 2x\geq -5[/tex]              .... (1)

[tex]5\geq f(x)\geq -5[/tex]

The range of f(x) is [-5,5].

Subtract 3 from each side of inequality (1).

[tex]5-3\geq -5\cos 2x-3\geq -5-3[/tex]

[tex]2\geq g(x)\geq -8[/tex]

The range of g(x) is [-8,2].

The function shifts down 3 units, so the range changes from −5 to 5 in f(x) to −8 to 2 in g(x).

Therefore option 2 is correct.

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