Problem 12 - Algebra 2

Question

Fountains The height of a fountain's water stream can be modeled by a quadratic function. Suppose the water from a jet reaches a maximum height of 8 feet at a distance 1 foot away from the jet. Suppose a worker increases the water pressure so that the stream reaches a maximum height of 12.5 feet at a distance of 15 inches from the jet. The water now lands 3.75 feet from the jet. Write a new quadratic function for \(H(d) .\) How do the changes in \(h\) and \(k\) affect the shape of the graph?

Step-by-Step Solution

Answer

The new function is \( H(d) = -2(d - 1.25)^2 + 12.5 \). Changes in \( h \) and \( k \) indicate a shift and increase in height.

Step 1: Understand the Quadratic Function

The quadratic function describing the height of the fountain's water stream is usually in vertex form: \( H(d) = a(d - h)^2 + k \), where \((h, k)\) is the vertex of the parabola. The vertex \( (h, k) \) represents the maximum height, and \( a \) affects the width and direction of the parabola.

Step 2: Identify the Vertex

The problem states that the maximum height of the water stream after increasing the pressure is 12.5 feet, reached at 15 inches from the jet. Convert 15 inches to feet by dividing by 12: \(\frac{15}{12} = 1.25\, \text{feet}.\) So, the vertex \((h, k)\) is \((1.25, 12.5)\).

Step 3: Establish the Coefficient 'a' Using Landing Point

The water lands 3.75 feet from the jet, so \( H(3.75) = 0 \). Substituting \( d = 3.75 \) and \( H(d) = 0 \) into the equation \( H(d) = a(d - 1.25)^2 + 12.5 \), we get \( 0 = a(3.75 - 1.25)^2 + 12.5 \). Solve for \( a \).

Step 4: Solve for 'a'

Calculate: \( 0 = a(2.5)^2 + 12.5 \). Therefore, \( 0 = 6.25a + 12.5 \). Solving for \( a \), we get \( a = -2 \).

Step 5: Write the New Quadratic Function

With \( a = -2 \), \( h = 1.25 \), and \( k = 12.5 \), the function is: \( H(d) = -2(d - 1.25)^2 + 12.5 \).

Step 6: Analyze Changes in 'h' and 'k'

The change in \( h \) from 1 foot to 1.25 feet means the maximum height is reached further from the jet. The change in \( k \) from 8 feet to 12.5 feet indicates a higher maximum height. The increased absolute value of \( a \) indicates the parabola is narrower.

Key Concepts

Vertex Form

When working with quadratic functions, one common representation is the vertex form, which makes it easy to identify the key characteristics of the function. The vertex form of a quadratic equation is expressed as:\[ H(d) = a(d - h)^2 + k \]In this formula:

• \( a \) controls the direction and "width" or "narrowness" of the parabola.
• \((h, k)\) represents the vertex, which is the highest or lowest point of the parabola.

For example, if a fountain's water stream follows this quadratic pattern, the vertex indicates the maximum height and its location relative to the jet. Here, \( h \) is the horizontal distance, and \( k \) is the maximum height above the ground.

Parabola

A parabola is the mathematical shape of the graph of a quadratic function. It has a curved "U" shape, and can open upwards or downwards depending on the coefficient \( a \).

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), the parabola opens downwards.

In the context of the fountain's water stream, we have a downward opening parabola because the water shoots upwards and then falls back down. The vertex, being the peak of the stream, provides the maximum height that the water reaches. Parabolas have axes of symmetry that run through their vertices, dividing them into two mirror-image halves. Understanding the shape and direction of the parabola helps in predicting the path of the stream.

Maximum Height

The maximum height in a quadratic function is the highest point the parabola reaches. It is determined by the vertex \((h, k)\), where \( k \) is the maximum height. In real-world applications like fountains, it shows how high the water will rise.For example, if a fountain's maximum height changes due to increased water pressure, it affects \( k \). Originally at 8 feet, if the pressure increases, the new maximum height might reach 12.5 feet. This change means the parabola shifts upwards, reflecting a new, higher point for the vertex. Understanding this relationship is crucial for applications requiring precise control of height, like designing fountain displays or ensuring safe water levels.

Coefficient

Coefficients are the numerical factors in mathematical equations, affecting the parabola's properties. In the vertex form of a quadratic function, the coefficient \( a \) plays a key role. It influences:

• The direction in which the parabola opens (upwards or downwards).
• The "width" or "narrowness" of the parabola.

A larger absolute value of \( a \), like \(-2\), makes the parabola narrower, meaning the water stream reaches its peak height more sharply. Conversely, a smaller absolute value would create a wider parabola, indicating a gentler peak. Thus, adjusting \( a \) directly impacts how quickly the maximum height is reached and how steeply the water returns to the ground. This detail can be essential in customizing water features for both aesthetic and functional purposes.