- The zeros are found by setting \(h(r) = 0\), which results in \(-(r+9)^2 = -36\), so \((r+9)^2 = 36\).
- Solving \(r+9 = \pm 6\) gives the smaller zero \(r = -15\) and the larger zero \(r = -3\).
- The vertex form \(h(r) = a(r-h)^2 + k\) allows us to identify the vertex directly as \((-9, 36)\).
- Zeros: \(r = -15, -3\); Vertex: \((-9, 36)\).
Explanation
- Analysis of the function
The function is given by \(h(r) = -(r + 9)^2 + 36\). We are asked to find the zeros of this function and identify the vertex of its corresponding parabola. A zero of a function is a value of \(r\) such that \(h(r) = 0\). The vertex form of a parabola is given by \(f(r) = a(r - h)^2 + k\), where \((h, k)\) represents the coordinates of the vertex.
- Finding the zeros of the function
To find the zeros of \(h(r) = -(r + 9)^2 + 36\), we set the function equal to zero:
\[-(r + 9)^2 + 36 = 0\]
Adding \((r + 9)^2\) to both sides, we get:
\[(r + 9)^2 = 36\]
Taking the square root of both sides, we obtain two possible equations for \(r\):
\[r + 9 = 6 \quad \text{or} \quad r + 9 = -6\]
Solving the first equation:
\[r = 6 - 9 = -3\]
Solving the second equation:
\[r = -6 - 9 = -15\]
Comparing the values, the smaller zero is \(-15\) and the larger zero is \(-3\).
- Identifying the vertex of the parabola
The function \(h(r) = -(r + 9)^2 + 36\) is already in the vertex form \(h(r) = a(r - h)^2 + k\). By comparing this to the standard vertex form, we can identify the parameters:
\(a = -1\)
\(h = -9\)
\(k = 36\)
The vertex is located at the coordinates \((h, k) = (-9, 36)\). This point represents the highest point on the parabola since the coefficient \(a = -1\) is negative, meaning the parabola opens downwards.
Examples
The study of quadratic functions like \(h(r) = -(r + 9)^2 + 36\) is essential in physics and engineering. For instance, if \(h(r)\) represents the height of a projectile at a distance \(r\) from a starting point, the zeros \(-15\) and \(-3\) would represent the points where the projectile hits the ground, and the vertex \((-9, 36)\) would tell us the maximum height reached by the projectile and the horizontal position at which that maximum height occurs. Understanding these properties helps in predicting flight paths and optimizing designs for structures or equipment.