Question

PROBLEM SOLVING A portion of the path that a hummingbird flies while feeding can be modeled by the function $$ f(x)=-\frac{1}{5} x(x-4)^2(x-7), 0 \leq x \leq 7 $$ where \(x\) is the horizontal distance (in meters) and \(f(x)\) is the height (in meters). The hummingbird feeds each time it is at ground level. a. At what distances does the hummingbird feed? b. A second hummingbird feeds 2 meters farther away than the first hummingbird and flies twice as high. Write a function to model the path of the second hummingbird.

Short answer

a. The first hummingbird feeds at the distances 0 meters, 4 meters, and 7 meters from the starting point. b. The function that describes the path of the second hummingbird is \(g(x) = -2/5 * (x-2)(x^2 - 8x + 16)^2 * (x - 9)\)

Step-by-step solution

Find values of x where hummingbird feeds

From the information given, a hummingbird feeds each time it's at the ground level. This means finding the values of \(x\) where \(f(x) = 0\). Therefore, solve the equation \(-\frac{1}{5}x(x-4)^2(x-7) = 0\) for \(x\). The solution set will provide the distances at which the first hummingbird feeds.

Solve the equation.

Equating the function to zero to find the roots, we get \(x=0, x=4\) and \(x=7\). So the hummingbird feeds at 0 meters, 4 meters, and 7 meters.

Write a function for the second hummingbird

According to the problem, the second hummingbird flies twice as high, and feeds 2 meters farther away than the first hummingbird. This suggests two transformations of the function \(f(x)\) for the second hummingbird. First, since the second hummingbird flies twice as high, multiply the function by 2. Second, because the second hummingbird feeds 2 meters farther away, replace \(x\) in the function with \(x-2\). This gives us the function for the flight of the second hummingbird as \( g(x) = 2f(x-2) = -2/5 * (x-2)((x-2) - 4)^2 * ((x-2) - 7)\).

Simplify the function for the second hummingbird

Expand and simplify the equation to \(g(x) = -2/5 * (x-2)(x^2 - 8x + 16)^2 * (x - 9)\)

Key concepts

Finding Zeros of Polynomial Functions

Understanding how to find the zeros of polynomial functions is essential when modeling real-world scenarios, such as the flight pattern of a hummingbird. Zeros of a polynomial function are the values of x that make the function value f(x) equal to zero. In simpler terms, these are the points where the graph of the function crosses or touches the x-axis.

In our exercise, the zeros are found by setting the polynomial function equal to zero and solving for x. This yields the values where the hummingbird reaches ground level, which translates to the feeding points. The polynomial given was factored, making it easy to see that the zeros occur at x = 0, x = 4, and x = 7. These points are crucial for understanding the behavior of the function and the hummingbird's flight.

Quadratic Transformations

Quadratic transformations involve changing a function's graph in various ways, including stretching or shrinking it vertically, flipping it over the x-axis or y-axis, and shifting it horizontally or vertically. Identifying these transformations allows us to model variations in behavior, such as when comparing two similar entities with minor differences in their patterns.

In the exercise about the two hummingbirds, we encounter transformations applied to the original function to model the flight of the second hummingbird. Multiplying the entire function by 2 indicates a vertical stretch; the second hummingbird flies 'twice as high'. Shifting the function horizontally is done by replacing every instance of x with (x - 2), which means the second hummingbird's feeding points are '2 meters farther away' than the first's. These transformations directly affect the graph and are reflected in the mathematical representation of the second bird's flight path.

Mathematical Modeling

Mathematical modeling uses mathematical expressions to represent real-world phenomena. By translating observed data into functions, we create models that can predict behavior under specific conditions. It's like crafting a mathematical story to describe what we see around us.

In our context, the polynomial function used to model the flight path of a hummingbird is a practical example of mathematical modeling. The function captures the essence of the bird's motion, feeding patterns, and how its flight path varies with distance. When the function is graphed, we can visualize the hummingbird's movement through space. This connection between algebra and geometry is what makes polynomial functions so powerful in modeling.

Graphing Polynomial Functions

Graphing polynomial functions involves plotting the shape of the polynomial based on its algebraic expression on a coordinate system. The graph helps us visualize the behavior of the function, which is particularly useful in understanding complex movements or patterns, as seen in the flight path of our hummingbirds.

By analyzing the graph, one can spot features such as intercepts, turning points, and the overall shape of the function's curve. For the hummingbirds' polynomial functions, the graph will reveal at which points the birds touch down to feed and how high they fly in between these points. The polynomial's degree gives us an idea of how many turns the graph will have, and its leading coefficient lets us know if the function opens upwards or downwards. Recognizing these features on the graph aids in interpreting the function and consequently, the real-world situation it represents.