1.) The cost in C dollars of manufacturing x bicycles at Holliday's Production Plant is given by the function C(x) = 2x2 - 800x + 92,000. Find the minimum cost.

2.) The vertex of y = -3x2- 6x - 9 lies in which quadrant?

3.) Given f(x) = 1/3*(x - 4 )2 + 7.
Identify the vertex.

4.) Write f(x) = -x2 - 2x + 3 in vertex form.

5.) Factor x2-13x + 42.

6.) Factor 6x2+13x + 6.

7.) Factor completely 4x3 - 81x.

the max or min of a quadratic is at the vertex of the parabola. I expect you can determine whether the vertex is a max or a min.

So, since the vertex is at x = -b/2a,

(2) -3x^2-6x-9
the vertex is at x = -6/(2*-3) = 1
y(1) = -18, so the vertex is at (1,-18)

(3) the vertex of y-k = a(x-h)^2 is at (h,k), so
the vertex is at (4,7)

(4) -x^2-2x+3
= -(x^2+2x+1)+3+1
= -(x+1)^2 + 4

(5) note that 42 = 6*7

(6) the discriminant is 5, so
x = (-13±5)/12 = -3/2, -2/3
(3x+2)(2x+3)

(7) 4x^3 - 81x
= x(4x^2 - 81)
= x(2x+9)(2x-9)

1.) To find the minimum cost, we need to locate the vertex of the given quadratic function. The vertex of any quadratic function in the form f(x) = ax^2 + bx + c can be found using the formula x = -b/ (2a).

In this case, the given function is C(x) = 2x^2 - 800x + 92,000. Comparing this with the general form, we have a = 2, b = -800, and c = 92,000.

Using the formula x = -b/ (2a), we can substitute the values:
x = -(-800) / (2*2) = 800/4 = 200.

Therefore, the minimum cost occurs when x = 200.

To find the minimum cost, substitute this value of x back into the original equation:
C(200) = 2(200)^2 - 800(200) + 92,000 = 80,000.

Hence, the minimum cost is $80,000.

2.) To determine which quadrant the vertex lies in, we need to find the x-coordinate of the vertex using the formula x = -b/ (2a).

In the given quadratic function, y = -3x^2 - 6x - 9, we can compare it with the general form to identify the values: a = -3 and b = -6.

Using the formula x = -b/ (2a), we can substitute these values:
x = -(-6) / (2*(-3)) = 6 / 6 = 1.

As the x-coordinate of the vertex is positive, the vertex lies in the positive x-axis. The y-coordinate of the vertex does not affect the quadrant.

Therefore, the vertex of the given quadratic lies in the first quadrant.

3.) To identify the vertex of the given function f(x) = 1/3(x - 4)^2 + 7, we can use the vertex form of a quadratic equation: f(x) = a(x - h)^2 + k. The vertex is represented by the coordinates (h, k).

From the given equation, we can identify that the vertex form is f(x) = 1/3(x - 4)^2 + 7. Comparing this with the standard form, we have h = 4 and k = 7.

Hence, the vertex of the given function is (4, 7).

4.) To write the function f(x) = -x^2 - 2x + 3 in vertex form, we need to complete the square.

First, rearrange the equation as follows: f(x) = -(x^2 + 2x) + 3.

Next, focus on the square term (x^2 + 2x). To complete the square, take half of the coefficient of the x-term (2) and square it (2/2)^2 = 1.

Now, add and subtract 1 inside the brackets: f(x) = -(x^2 + 2x + 1 - 1) + 3.

Rearrange the equation and factor the quadratic expression inside the brackets: f(x) = -(x^2 + 2x + 1) + 1 + 3.

Simplify: f(x) = -((x + 1)^2 - 1) + 4.

Distribute the negative sign: f(x) = -(x + 1)^2 + 1 + 4.

Combine the constants: f(x) = -(x + 1)^2 + 5.

Hence, the given function f(x) = -x^2 - 2x + 3 in vertex form is f(x) = -(x + 1)^2 + 5.

5.) To factor the expression x^2 - 13x + 42, we need to find two numbers that multiply to give the constant term (42) and add to give the coefficient of the middle term (-13).

In this case, the two numbers are -6 and -7. Let's factor the expression accordingly:

x^2 - 13x + 42 = (x - 6)(x - 7).

Hence, the factored form of x^2 - 13x + 42 is (x - 6)(x - 7).

6.) To factor the expression 6x^2 + 13x + 6, we need to find two numbers that multiply to give the constant term (6) and add to give the coefficient of the middle term (13).

In this case, the two numbers are 2 and 3. Let's factor the expression accordingly:

6x^2 + 13x + 6 = (2x + 3)(3x + 2).

Hence, the factored form of 6x^2 + 13x + 6 is (2x + 3)(3x + 2).

7.) To factor completely the expression 4x^3 - 81x, we can identify the common factor of the expression, which is x. Factoring out x gives:

4x^3 - 81x = x(4x^2 - 81).

Now, focus on the second term, 4x^2 - 81, which is a difference of squares. It can be factored as follows:

4x^2 - 81 = (2x)^2 - 9^2.

Applying the difference of squares formula, we get:

(2x - 9)(2x + 9).

Therefore, the expression 4x^3 - 81x can be factored completely as:

x(2x - 9)(2x + 9).

1.) To find the minimum cost, we need to find the vertex of the function C(x) = 2x^2 - 800x + 92,000. The x-coordinate of the vertex is given by the formula x = -b / (2a), where a and b are coefficients of the quadratic equation. In this case, a = 2 and b = -800.

x = -(-800) / (2 * 2) = 800 / 4 = 200.

To find the corresponding y-coordinate, substitute x = 200 into the function:

C(200) = 2(200)^2 - 800(200) + 92,000 = 80,000 - 160,000 + 92,000 = 12,000.

Therefore, the minimum cost is $12,000.

2.) The vertex form of a quadratic equation is given by y = a(x - h)^2 + k, where (h, k) represents the coordinates of the vertex. In this case, the equation is y = -3x^2 - 6x - 9.

To find the x-coordinate of the vertex, use the formula x = -b / (2a), where a = -3 and b = -6.

x = -(-6) / (2 * -3) = 6 / -6 = -1.

Substitute x = -1 into the equation to find the y-coordinate:

y = -3(-1)^2 - 6(-1) - 9 = -3 + 6 - 9 = -6.

The vertex is (-1, -6). To determine the quadrant it lies in, we will consider the sign of a and the shape of the parabola. In this case, a is negative (-3), so the parabola opens downwards. Thus, the vertex (-1, -6) lies in the third quadrant.

3.) To identify the vertex, we can rewrite the equation f(x) = 1/3(x - 4)^2 + 7 in the vertex form, y = a(x - h)^2 + k. Comparing the two forms, we can determine that the vertex is at point (h, k). In this case, the vertex is (4, 7).

4.) To write the equation f(x) = -x^2 - 2x + 3 in vertex form, we need to complete the square.

Start by factoring out the coefficient of x^2:

f(x) = -(x^2 + 2x) + 3.

Next, complete the square by adding and subtracting the square of half the coefficient of x:

f(x) = -(x^2 + 2x + 1 - 1) + 3.

Now, let's rewrite the equation:

f(x) = -[(x + 1)^2 - 1] + 3.

Using the distributive property:

f(x) = -(x + 1)^2 + 1 + 3.

Simplifying further:

f(x) = -(x + 1)^2 + 4.

Therefore, the equation in vertex form is f(x) = -(x + 1)^2 + 4.

5.) To factor the quadratic equation x^2 - 13x + 42, we need to find two numbers that multiply to give 42 and add up to -13. These numbers are -6 and -7.

Therefore, we can write the quadratic equation as:

(x - 6)(x - 7).

6.) To factor the quadratic equation 6x^2 + 13x + 6, we need to find two numbers that multiply to give 6 * 6 = 36 and add up to 13. These numbers are 4 and 9.

Therefore, we can write the quadratic equation as:

(3x + 2)(2x + 3).

7.) To factor completely the equation 4x^3 - 81x, we factor out the greatest common factor, which is x:

x(4x^2 - 81).

The expression (4x^2 - 81) is a difference of squares, so we can factor it further as:

x(2x + 9)(2x - 9).

Therefore, the equation 4x^3 - 81x can be factored completely as x(2x + 9)(2x - 9).