Find the minimum or maximum value of each function. then State the domain and range of the function. 1. g(x)=x^2-2x+1 2.h(x)= 5x^2+ 15x-3

1.

The minimum value of g(x) is (1 , 0)

The domain of g(x) is {x : x ∈ R}

The range of g(x) is {y : y ≥ 0}

2.

The minimum value of h(x) is (-1.5 , -14.25)

The domain of h(x) is {x : x ∈ R}

The range of h(x) is {y : y ≥ -14.25}

Step-by-step explanation:

The quadratic function f(x) = ax² + bx + c is represented graphically by a parabola which has a minimum/maximum vertex (h , k), where

h = and k is the value of f(h)The minimum/maximum value of the function is (h , k)The domain of the function is {x : x ∈ R} (all real numbers)The range of the function is {y : y ≥ k} if the function has a minimum value OR {y : y ≤ k} if the function has a maximum valueIf a (coordinate x²) is positive, then the function has a minimum value  if a is negative, then the function has a maximum value

Now let us solve the problems

1.

∵ g(x) = x² - 2x + 1

∵ The form of the quadratic function is f(x) = ax² + bx + c

∴ a = 1 , b = -2 , c = 1

∵ a is positive

∴ The function has a minimum value

∵ h =

∴ h =

- To find k substitute x in g(x) by 1

∴ g(1) = (1)² - 2(1) + 1 = 1 - 2 + 1

∴ g(1) = 0

∵ k = g(h)

∴ k = 0

∴ The minimum value of g(x) is (1 , 0)

∵ The domain of the quadratic function is all real numbers

∴ The domain of g(x) is {x : x ∈ R}

∵ The range of the quadratic function with minimum value is

   {y : y ≥ k}

∴ The range of g(x) is {y : y ≥ 0}

2.

∵ h(x) = 5x² + 15x - 3

∴ a = 5 , b = 15 , c = -3

∵ a is positive

∴ The function has a minimum value

∵ h =

∴ h =

∴ h = - 1.5

- To find k substitute x in h(x) by 1

∴ h(-1.5) = 5(-1.5)² + 15 (-1.5) - 3 = 11.25 - 22.5 - 3

∴ h(-1.5) = -14.25

∵ k = h(h)

∴ k = -14.25

∴ The minimum value of h(x) is (-1.5 , -14.25)

∵ The domain of the quadratic function is all real numbers

∴ The domain of h(x) is {x : x ∈ R}

∵ The range of the quadratic function with minimum value is

   {y : y ≥ k}

∴ The range of h(x) is {y : y ≥ -14.25}

The Millers paddled 4,666 miles each day.  If they reduced their daily distance by half, extending their trip to twice as many days, the distance traveled by the Millers would have been the same.

Step-by-step explanation:

Given that the Miller family likes to paddle along the river, and they paddled the same distance each day throughout the course of three days, traveling a total of 14 miles, to determine how many miles did they travel each day the following calculation must be performed :

14/3 = X

4.666 = X

Thus, the Millers paddled 4,666 miles each day.

In turn, if the Millers went half their daily distance each day, but extend their trip to twice as many days, to determine how far would they travel, the following calculation must be performed:

4.666 / 2 x 6 = X

2.333 x 6 = 13.9999

Thus, the distance traveled by the Millers would have been the same.