A trigonometric function is defined as f(x) = a \sin(b x) + c, where a > 0 and b > 0. The … — JAMB UTME

A trigonometric function is defined as $f(x) = a \sin(b x) + c$ , where $a > 0$ and $b > 0$ . The graph of $y = f(x)$ has a maximum value of $8$ , a minimum value of $-2$ , and a period of $120^\circ$ . What is the value of $a + b + c$ ?

Answer: 11

Answer

The value of

a + b + c

is 11.

Using the maximum (

a + c = 8

) and minimum (

-a + c = -2

), solving simultaneously gives

a = 5

and

c = 3

. Using the period formula

T = \frac{360^\circ}{b} = 120^\circ

, we obtain

b = 3

. Adding these values together yields

5 + 3 + 3 = 11

Step-by-Step Solution

Set up a system of linear equations for the amplitude and vertical shift from the maximum and minimum bounds

a = 5

and

c = 3

Since

\sin(bx)

ranges from

-1

1

, the maximum is

a(1) + c = 8

and minimum is

a(-1) + c = -2

. Solving these simultaneous equations gives

c = 3

and

a = 5

Calculate the frequency coefficient

b

using the period formula

b = 3

For a sine curve specified in degrees, the period is

T = \frac{360^\circ}{b}

. Substituting

T = 120^\circ

yields

b = 3

Compute the requested sum

a + b + c

a + b + c = 11

Summing the calculated parameters

5 + 3 + 3 = 11

Key Concept

Determining parameters of a trigonometric graph from amplitude, vertical shift, and period.

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