Question

MODELING WITH MATHEMATICS Flying fi sh use their pectoral fi ns like airplane wings to glide through the air. a. Write an equation of the form y = a(x ? h)2 + k with vertex (33, 5) that models the fl ight path, assuming the fi sh leaves the water at (0, 0). b. What are the domain and range of the function? What do they represent in this situation? c. Does the value of a change when the fl ight path has vertex (30, 4)? Justify your answer.

Short answer

The equation of the flight path is \(y = 0.0046(x - 33)^2 + 5\), with domain \([0, 33]\) and range \([0, 5]\). The value of a changes when the flight path has vertex (30, 4), as it becomes \(0.0044\).

Step-by-step solution

Write the equation of the flight path

To write the equation of the flight path, we will use the vertex form \(y = a(x - h)^2 + k\). The given vertex is (33, 5). So, substitute \(h = 33\) and \(k = 5\) into the formula. We also know that the fish leaves the water at (0, 0). So, substitute \(x = 0\) and \(y = 0\) into the formula and solve for \(a\). The resulting equation is \(y = 0.0046(x - 33)^2 + 5\).

Determine the domain and range of the function

The domain of the function is the set of all possible x-values. In this situation, since the fish leaves the water at \(x = 0\) and its highest point is at \(x = 33\), the domain is \([0, 33]\). The range of the function is the set of possible y-values. In this situation, since the fish leaves the water at \(y = 0\) and its highest point is \(y = 5\), the range is \([0, 5]\). The domain represents the distance the fish travels, while the range represents the height of the fish's flight.

Assess the change in value of \(a\) given a new vertex

If the flight path has a new vertex (30, 4), the equation of the flight path becomes \(y = a(x - 30)^2 + 4\). Solving for \(a\) using the point (0, 0) gives \(a = 0.0044\). However, this is not equal to the original \(a = 0.0046\), so the value of \(a\) does change when the flight path has a different vertex.

Key concepts

Vertex Form

When working with quadratic equations, one of the most useful representations is the vertex form. The vertex form of a quadratic equation is given by \(y = a(x - h)^2 + k\). This format is particularly valuable as it directly reveals the vertex of the parabola, which is \((h, k)\). The vertex represents the highest or lowest point on the parabola, depending on the value of \(a\).

For the flight of the fish in the exercise, the vertex form helps us capture the peak point of the fish’s trajectory. By using the vertex \((33, 5)\), we find that \(h = 33\) and \(k = 5\). To construct the equation for the flight path, we also consider that the fish starts its glide from \((0, 0)\).

By substituting these values into the vertex form equation and solving for \(a\), which determines the "width" and "direction" of the parabola, the final equation representing the flight is \(y = 0.0046(x - 33)^2 + 5\). This describes a parabola opening upwards with its vertex as the highest point, modeling the parabolic flight path accurately.

Domain and Range

Understanding the domain and range of a function is crucial when interpreting real-world situations using mathematics. The **domain** refers to all the possible values that \(x\) can take, while the **range** pertains to the values that \(y\) can achieve.

In the case of the fish's flight path, the domain reflects the horizontal distance the fish travels. Starting from the water at \(x = 0\) and reaching its peak distance at \(x = 33\), the domain is \[ [0, 33] \]. This means the horizontal stretch of the fish's flight in the air is from 0 to 33 units.

The range, on the other hand, captures the vertical reach or the height of the fish's trajectory. Since the fish starts at \(y = 0\) and reaches a maximum height of \(y = 5\), the range is \[ [0, 5] \]. This height variance represents how high the fish gets above the water during its glide. Understanding these two aspects allows for a comprehensive view of the pathway.

• The domain signifies the horizontal boundaries of movement.
• The range articulates the vertical extent of the path.

Parabolic Path

A parabolic path is a common trajectory described by a quadratic equation, particularly evident in natural and mechanical phenomena. The trajectory is symmetrical and defined by its leading coefficient \(a\) and vertex \((h, k)\).

In modeling the fish's flight path, the parabolic curve gives insight into the behavior of the motion. The fish's jump takes a symmetrical arc due to the quadratic relationship modeled in the equation. The **coefficient** \(a\) plays a critical role; it not only affects how "tall" and "narrow" the parabola is but also how "fast" the fish's trajectory changes direction.

When the initial vertex changes from \((33, 5)\) to \((30, 4)\), recalculating \(a\) with the point \((0, 0)\) gives a new \(a = 0.0044\), slightly altering the shape and path of the parabola. This shift indicates a slightly different curve, affecting the fish's glide in terms of distance and height.

• The shape and direction of a parabolic path are influenced by \(a\).
• Modifying the vertex coordinates will distinctly alter \(a\).

These changes demonstrate how sensitive parabolic paths are to their parameters, signifying adjustments in physical phenomena like the fish's flight.