40 points functions f(x) and g(x) are shown below: f(x) = 3x2 + 12x + 16 g(x)= 2 *sin(x-pi) using complete sentences, explain how to find the minimum value for each function and determine which function has the smallest minimum y-value.

the function g(x) has the smallest minimum y-value.

Explanation:

1) The function f(x) = 3x² + 12x + 16 is a parabola.

The vertex of the parabola is the minimum or maximum on the parabola.

If the parabola open down then the vertex is a maximum, and if the parabola open upward the vertex is a minimum.

The sign of the coefficient of the quadratic term tells whether the parabola opens upward or downward.

When such coefficient is positive, the parabola opens upward (so it has a minimum); when the coefficient is negative the parabola opens downward (so it has a maximum).

Here the coefficient is positive (3), which tells that the vertex of the parabola is a miimum.

Then, finding the minimum value of the function is done by finding the vertex.

I will change the form of the function to the vertex form by completing squares:

Given: 3x² + 12x + 16

Group: (3x² + 12x) + 16
Common factor: 3 [x² + 4x ] + 16
Complete squares: 3[ ( x² + 4x + 4) - 4] + 16
Factor the trinomial: 3 [(x + 2)² - 4] + 16
Distributive property: 3 (x + 2)² - 12 + 16
Combine like terms: 3 (x + 2)² + 4

That is the vertex form: A(x - h)² + k, whch means that the vertex is (h,k) = (-2, 4).

Then the minimum value is 4 (when x = - 2).

2) The othe function is g(x)= 2 *sin(x-pi)


The sine function goes from -1 to + 1, so the minimum value of sin(x - pi) is - 1.

When you multiply by 2, you just increased the amplitude of the function and obtain the new minimum value is 2 (-1) = - 2

Comparing the two minima, you have 4 vs - 2, and so the function g(x) has the smallest minimum y-value.