Question

Functions f(x) and g(x) are shown below:

f(x) g(x)
f(x) = 3x2 + 12x + 16 graph of sine function which starts at 0 comma 0 and decreases to the minimum pi over 2, then increases to the maximum of 3 pi over 2 then decreases to 2 pi where the cycle repeats
Courtesy of Texas Instruments


Using complete sentences, explain how to find the minimum value for each function and determine which function has the smallest minimum y-value.

Answer 1

g(x) has the smallest minimum y-value

Explanation:

f(x) is the equation of a parabola

The general form of a parabola is ax² + bx + c, if a is positive, the parabola has a minimum. The minimum is at the vertex, the x-coordinate of the vertex is calculated as follows: -b/2a. For this case, x =-12/(2*3) = -2, which corresponds to the following function value: 3(-2)² + 12(-2) + 16 = 4

• minimum value of f(x) = 4

g(x) is the sine function

The sine function is a periodic function which oscillates between -1 and 1

• minimum value of g(x) = -1

Answer 2

f` ( x ) = 6 x + 12
6 x + 12 = 0 6 x = - 12 x = - 2 f ( - 2 ) 0 12 - 24 + 16 = 4 f ( x ) min = 4 g` ( x ) = 4 cos ( 2 x - π ) 4 cos ( 2 x - π ) = 0 cos ( 2 x - π ) = 0 2 x - π = 3π / 2 2 x = 5π /2 x = 5π/4 g ( 5π/4 ) = 2 sin ( 5π/2 - π ) + 4 = 2 ( sin 3π/2 ) + 4 = -2 + 4 = 2 g ( x ) min = 2