To solve this problem, let's break down each part.

Part (a): Describe the graph of $f(x) = 0.00000093x^2 - 0.0145x + 60$

The function $f(x)$ is a quadratic function of the form: $f(x) = ax^2 + bx + c$ where:

• $a = 0.00000093$
• $b = -0.0145$
• $c = 60$

In a quadratic function, the sign of the $a$-coefficient determines whether the parabola opens upward or downward:

• If $a > 0$, the parabola opens upward.
• If $a < 0$, the parabola opens downward.

Since $a = 0.00000093$, which is positive, the parabola opens upward. Therefore, the correct answer is:

• A. a parabola that opens upward

Part (b): Find the minimum average cost per copy and the corresponding number of copies made.

For a parabola that opens upward, the minimum point (vertex) of the function represents the lowest point on the graph. To find the x-coordinate of the vertex for a quadratic function $f(x) = ax^2 + bx + c$, we use the formula: $x = -\frac{b}{2a}$

Substituting $a = 0.00000093$ and $b = -0.0145$: $x = -\frac{-0.0145}{2 \times 0.00000093} = \frac{0.0145}{0.00000186} \approx 7795.70$

The minimum average cost per copy occurs when approximately $x = 7796$ copies are produced. Now, let's substitute $x = 7796$ back into the function $f(x)$ to find the minimum average cost:

$f(7796) = 0.00000093(7796)^2 - 0.0145 \times 7796 + 60$ Calculating each term:

1. $0.00000093 \times 7796^2 \approx 56.50$
2. $-0.0145 \times 7796 \approx -113.04$
3. Constant term $+ 60$

Adding these up: $f(7796) \approx 56.50 - 113.04 + 60 \approx 3.46$

So, the minimum average cost per copy is approximately 3.46 cents, and this occurs when 7796 copies are made.

Would you like further details on any part of the solution?

Related Questions:

1. How would the answer change if the coefficient $a$ were negative?
2. Can you explain how to find the maximum point of a downward-opening parabola?
3. What does the minimum cost represent in terms of the business's operation?
4. How does changing the values of $a$, $b$, or $c$ affect the shape and position of the graph?
5. How can the vertex formula be derived from completing the square?

Tip: For optimization problems involving quadratic functions, always check if the parabola opens upward or downward to know whether you are finding a minimum or maximum.