Problem 46 - Algebra 1: Concepts and Skills

Question

In Exercises 46 and 47 use the following information. A bottle-nosed dolphin jumps out of the water. The path the dolphin travels can be modeled by \(h=-0.2 d^{2}+2 d,\) where \(h\) represents the height of the dolphin and \(d\) represents horizontal distance. What is the vertex of the parabola? Interpret the result.

Step-by-Step Solution

Answer

The vertex of the parabola is (10,5), suggesting that the dolphin reaches its maximum height of 5 meters at a horizontal distance of 10 meters from its starting point.

Step 1: Identify the Form of the Equation

Recognize that the function \(h = -0.2d^{2}+2d\) represents a parabola. The standard form of a parabolic function is \(f(x) = a(x-h)^{2} + k\), where (h, k) is the vertex of the parabola. However, this function is in the form \(f(x) = ax^{2} + bx\). To find the vertex, it is useful to first rewrite it into the vertex form

Step 2: Convert to Vertex Form

To convert the function to vertex form, one can complete the square. To do this, keep the coefficient of \(d^{2}\) and factor out that term (which would be -0.2) from both \(d^{2}\) and d. It should look like this: \( h = -0.2(d^{2}- (2/-0.2)d)\). Now, compute the square that needs to be added and subtracted to make a perfect square trinomial inside the parentheses. That is the square of half of the coefficient of d, \((-10/2)^{2} = 25\). So, the equation becomes \(h = -0.2((d-10)^{2} - 25)\). Open, we get \(h = -0.2(d - 10)^{2} + 5\)

Step 3: Find and Interpret the Vertex

Now, the function is in vertex form \(f(x) = a(x-h)^{2} + k\), where (h, k) is the vertex of the parabola. It is clear that the vertex is (10, 5). Now, interpreting this result in the context of the problem, the number 10 (h-coordinate of the vertex) represents the horizontal distance the dolphin travels when it reaches maximum height, and this distance is 10 (units depending on the unit of measurement used for distance, presumed to be meters). The number 5 (k-coordinate of the vertex) represents this maximum height, so the dolphin jumps 5 units high (again, presumed units are meters).

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial equation that is characterized by a squared term. It takes the general form:

• \( f(x) = ax^2 + bx + c \)

where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable.
This specific structure creates a graph known as a parabola, which can open upwards or downwards.
The coefficient \(a\) determines the direction and "width" of the parabola. If \(a\) is positive, the parabola opens upwards, and if \(a\) is negative, it opens downwards.
The quadratic function used in this dolphin problem is
\( h = -0.2d^2 + 2d \). It represents how the height \( h \) of the dolphin changes with the horizontal distance \( d \).
Notice the "negative" \(-0.2d^2\) which, given it's a negative number, indicates the parabola opens downward, which is typical when calculating maximum heights.

Parabolic Path

A parabolic path is a trajectory outlined by a parabola. It is common in real-world scenarios such as projectile motion.
The equation of the dolphin's jump defines a parabolic path since the trajectory it follows is determined by a quadratic function.
This path is symmetrical and its highest point, or vertex, is crucial to understanding properties like the maximum height the dolphin reaches.
Key characteristics of a parabolic path include:

• The path is symmetrical about a vertical line, known as the axis of symmetry.
• The path has one extreme point, the vertex, which is either a maximum or a minimum.

In this context, learning about the vertex helps us interpret the dolphin's jump, specifically how far horizontally (10 meters here) and how high it goes (5 meters here).

Completing the Square

Completing the square is a technique used to convert a quadratic equation into its vertex form.
This transformation makes it easier to identify the vertex of the parabola, which gives valuable insights into its graph like maximum or minimum points.
The original quadratic equation for the dolphin's path is
\( h = -0.2d^2 + 2d \). To complete the square:

• First, factor out the \(-0.2\) from the quadratic and linear term.
• Observe the transformation: \(h = -0.2(d^2 - 10d)\).
• Next, take half of the linear coefficient \(-10/2\), square it (25), and add and subtract it within the brackets: \(h = -0.2((d-10)^2 - 25)\).

This manipulates the equation into its vertex form \(h = -0.2(d - 10)^2 + 5\).
This form directly shows the vertex, in this case, the point \((10, 5)\), without needing to complete further calculations.

Maximum Height Interpretation

Interpreting the vertex of a parabola helps us find practical information like maximum or minimum values in a real-world context, such as the flight of a dolphin.
For the given dolphin jump, the vertex is at \((10, 5)\).
Here this means:

• The "10" represents the horizontal distance \(d\), indicating the dolphin travels 10 meters horizontally before reaching maximum height.
• The "5" is the maximum height \(h\) of the jump, indicating the dolphin reaches 5 meters above the water.

Understanding these interpretations allows us to visualize the jump, making it clear how high and how far the dolphin leaps.
Such insights provided by vertices are significant in fields like physics and engineering where calculating maximum reach or optimizing performances are essential.