Let's break down and solve the questions step-by-step based on the problem provided.

Problem Summary

The path of a hummingbird while feeding is modeled by the function: $F(x) = (x - 4)^2 - 1$ where $0 \leq x \leq 70$. Here, $x$ represents the horizontal distance (in meters), and $F(x)$ represents the height (in meters) above the ground.

Questions

1. Part a: At what distances does the hummingbird feed?

   • The hummingbird feeds when it is at ground level, meaning $F(x) = 0$.

2. Part b: A second hummingbird flies along a path that is translated vertically 3 meters higher. Write a function $g(x)$ to model the path of this second hummingbird.

3. Extension: What if a third hummingbird flies twice as high as $f(x)$? Determine the function modeling this path, describe the transformation, and explain how it affects the graph.

Solutions

Part (a): Finding Feeding Distances

The hummingbird feeds at ground level, which occurs when $F(x) = 0$.

1. Set the function equal to zero: $(x - 4)^2 - 1 = 0$

2. Solve for $x$: $(x - 4)^2 = 1$ Taking the square root of both sides, we get: $x - 4 = \pm 1$

3. Solve for $x$ in each case:

   • $x - 4 = 1 \Rightarrow x = 5$
   • $x - 4 = -1 \Rightarrow x = 3$

Thus, the hummingbird feeds at distances $x = 3$ meters and $x = 5$ meters.

Part (b): Function for Second Hummingbird Path

The second hummingbird’s path is translated vertically 3 meters higher. To achieve this, we add 3 to the original function $F(x)$.

1. Define $g(x)$ as: $g(x) = F(x) + 3 = (x - 4)^2 - 1 + 3$

2. Simplify: $g(x) = (x - 4)^2 + 2$

Therefore, the function for the second hummingbird's path is: $g(x) = (x - 4)^2 + 2$

Extension: Third Hummingbird Path

If a third hummingbird flies twice as high as $f(x)$, we scale the height by a factor of 2.

1. Define $h(x)$ as: $h(x) = 2 \cdot F(x) = 2 \cdot ((x - 4)^2 - 1)$

2. Expand and simplify: $h(x) = 2(x - 4)^2 - 2$

This is the function for the third hummingbird's path: $h(x) = 2(x - 4)^2 - 2$

Transformation Description:

• This transformation is a vertical stretch by a factor of 2.
• The graph will be more "narrow" or "taller" compared to the original path since every point is doubled in height.

Would you like further details on any part of this solution, or do you have additional questions?

Further Exploration Questions:

1. What would the graph of each function look like?
2. How does translating a function vertically impact its maximum and minimum values?
3. Can you find the maximum height the hummingbird reaches in each path?
4. What would happen if the second hummingbird's path were shifted 3 meters down instead of up?
5. How would a horizontal translation affect the function $F(x)$?

Tip

Remember that translating a function vertically involves adding or subtracting from the entire function, while horizontal shifts change the input variable $x$ inside the function.