An ornithologist studying the flight of birds measured the efficiency for parakeets flying at various speeds in a descending
flight pattern, 0.18 at 12 mph, 0.23 at 22 mph and 0.18 at 30 mph. The measurements suggest the efficiency E is a quadratic
function of the speed s. Find the quadratic function whose graph goes through the three given ordered pairs, and find the
speed that gives the maximum efficiency for descending flight.

...

The equation is y =
(Round to four decimal places as needed.)

Given that the efficiency \( E \) is a quadratic function of speed \( s \), we can express this as:

\[
E(s) = as^2 + bs + c
\]

We are given three points corresponding to the speed and efficiency values:

- \( (12, 0.18) \)
- \( (22, 0.23) \)
- \( (30, 0.18) \)

These points give us three equations based on the general quadratic formula:

1. \( 0.18 = a(12)^2 + b(12) + c \)
2. \( 0.23 = a(22)^2 + b(22) + c \)
3. \( 0.18 = a(30)^2 + b(30) + c \)

We will solve this system of equations to find the values of \( a \), \( b \), and \( c \), then find the speed \( s \) that gives the maximum efficiency by determining the vertex of the quadratic function.

Let's proceed with the calculations.The quadratic function representing the efficiency \( E(s) \) in terms of speed \( s \) is:

\[
E(s) = -0.000625s^2 + 0.02625s - 0.045
\]

To find the speed that gives the maximum efficiency, we need to determine the vertex of this quadratic function. The speed at the vertex of a parabola given by \( ax^2 + bx + c \) occurs at:

\[
s = -\frac{b}{2a}
\]

Let's calculate the speed \( s \) that maximizes the efficiency.The speed that gives the maximum efficiency for descending flight is \( 21 \) mph.

Would you like more details on how the quadratic function was derived or have any other questions?

Here are five related questions to explore further:
1. How does a quadratic function model different types of parabolas?
2. What is the significance of the vertex in real-world applications?
3. How do we determine the direction of the parabola's opening?
4. What role do the coefficients \( a \), \( b \), and \( c \) play in the shape of the graph?
5. How can we verify the correctness of the quadratic model using the given data points?

**Tip:** The vertex formula \( s = -\frac{b}{2a} \) is crucial in finding optimal values in quadratic models, such as maximizing or minimizing real-world quantities.

An ornithologist studying the flight of birds measured the efficiency for parakeets flying at various speeds in a descending flight pattern, 0.18 at 12 mph, 0.23 at 22 mph, and 0.18 at 30 mph. The measurements suggest the efficiency E is a quadratic function of the speed s. Find the quadratic function whose graph goes through the three given ordered pairs, and find the speed that gives the maximum efficiency for descending flight.

Discover how to derive a quadratic function to model the flight efficiency of parakeets at different speeds. This problem involves key algebraic concepts such as the quadratic equation and the vertex formula, making it ideal for students in Grades 10-12.

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