Question

PROBLEM SOLVING The path of a diver is modeled by the function f(x) = ?9x2 + 9x + 1, where f(x) is the height of the diver (in meters) above the water and x is the horizontal distance (in meters) from the end of the diving board. a. What is the height of the diving board? b. What is the maximum height of the diver? c. Describe where the diver is ascending and where the diver is descending.

Short answer

The diving board is at a height of 1 meter. The maximum height the diver reaches is 2.75 meters. The diver is ascending when x is between 0 and 0.5, and descending for x > 0.5.

Step-by-step solution

Identifying the Initial Height

The initial height of the diving board can be determined by finding the value of the function at x = 0. So, substituting x=0 into the function \(f(x) = - 9x^2 + 9x + 1\) gives \(f(0) = 1\). So, the diving board is at a height of 1 meter above the water.

Determining Maximum Height

The maximum height of the diver can be found by finding the vertex of the parabola. As the function is a downward-facing parabola, the vertex represents the maximum point. The x-coordinate of the vertex can be found using the formula \(-b / 2a\). Substituting a = -9 and b = 9 gives \(x = - (9) / (2 * -9) = 0.5\). Substituting x=0.5 in function yields \(f(0.5) = - 9(0.5^2) + 9(0.5) + 1 = 2.75\), so the maximum height of the diver is 2.75 meters.

Describing the Diver's Ascent and Descent

As the function represents a downward-facing parabola, the diver ascends from x = 0 to the x-coordinate of the vertex (x = 0.5). After reaching the vertex point, the diver begins descending. So, the diver is ascending when x is between 0 and 0.5, and descending for x > 0.5.

Key concepts

Vertex of a Parabola

The vertex of a parabola is a crucial point we need to understand when analyzing quadratic functions. A parabola can either open upwards or downward. In our case, the function \( f(x) = -9x^2 + 9x + 1 \) defines a downward-opening parabola. The vertex acts as the highest point for a downward parabola, representing the peak height in this scenario for the diver's path.
To find the vertex of a parabola, particularly the x-coordinate, we use the formula \( x = \frac{-b}{2a} \). Here, \( a \) and \( b \) are the coefficients from the quadratic equation \( ax^2 + bx + c \). By substituting \( a = -9 \) and \( b = 9 \) into the formula:

• \( x = \frac{-9}{2 \times -9} = 0.5 \)

This tells us that the vertex occurs at \( x = 0.5 \). Knowing this x-value allows us to determine the maximum height of the path, which we will explore further. The x-coordinate of the vertex gives us the point where ascent changes to descent in a parabolic path, a concept used to assess the diver’s motion.

Maximum Height

Maximum height is an essential consideration when analyzing quadratic functions, particularly for physical scenarios like our diver's path. As we determined earlier, the vertex of the parabola in our function represents this maximum height.
To find it, we substitute the vertex x-coordinate back into the function. So, for our function \( f(x) = -9x^2 + 9x + 1 \), substitute \( x = 0.5 \) to find the height:

• Calculate \( f(0.5) = -9(0.5)^2 + 9(0.5) + 1 \)
• \( f(0.5) = -9 \times 0.25 + 4.5 + 1 \)
• \( f(0.5) = -2.25 + 4.5 + 1 = 2.75 \)

Thus, the maximum height the diver reaches is 2.75 meters above the water. Evaluating this point provides critical insight into peak performance in motion paths, especially important for safety and competitive assessments in diving scenarios.

Ascent and Descent of Function

Understanding the ascent and descent segments of a quadratic function is necessary for comprehending the full motion path described by the function. The diver, described by the function \( f(x) = -9x^2 + 9x + 1 \), starts ascending from the diving board height at \( x = 0 \) until reaching the maximum height.
This ascent occurs between the horizontal distance \( x = 0 \) and the vertex at \( x = 0.5 \), where the diver reaches their peak. During this interval, the function values increase as \( x \) moves towards 0.5.

• Ascending: from \( x = 0 \) to \( x = 0.5 \)

Once the diver passes the vertex \( x = 0.5 \), they begin their descent until they reach the water. Descent in the graph corresponds to the decreasing function values beyond this peak.

• Descending: when \( x > 0.5 \)

Visualizing the ascent and descent aligns with understanding real-world applications of parabolas, making these complex mathematical concepts more relatable.