Questions: Use your graphing calculator to graph the function n(r)=-r^2-19r-25. Change the window for both the x-coordinate and the y-coordinate in order to get the relative maximum of the parabola on the same screen. Find the coordinates of the relative maximum. Round to the nearest hundredth. The relative maximum is

Use your graphing calculator to graph the function n(r)=-r^2-19r-25. Change the window for both the x-coordinate and the y-coordinate in order to get the relative maximum of the parabola on the same screen. Find the coordinates of the relative maximum. Round to the nearest hundredth.

The relative maximum is

Transcript text: Use your graphing calculator to graph the function $n(r)=-r^{2}-19 r-25$. Change the window for both the $x$-coordinate and the $y$-coordinate in order to get the relative maximum of the parabola on the same screen. Find the coordinates of the relative maximum. Round to the nearest hundredth. The relative maximum is $\square$

Solution

Solution Steps

To find the relative maximum of the quadratic function $ n(r) = -r^2 - 19r - 25 $, we can use calculus or the vertex formula for a parabola. Since the parabola opens downwards (the coefficient of $ r^2 $ is negative), the vertex will give us the relative maximum. The vertex of a parabola $ ax^2 + bx + c $ is at $ x = -\frac{b}{2a} $. We can substitute this value back into the function to find the maximum value.

Step 1: Identify the Function

The function given is $ n(r) = -r^2 - 19r - 25 $.

Step 2: Determine the Vertex

For a quadratic function $ ax^2 + bx + c $, the vertex $ r $ is found using the formula: \[ r = -\frac{b}{2a} \] Here, $ a = -1 $ and $ b = -19 $. Substituting these values, we get: \[ r = -\frac{-19}{2 \times -1} = -\frac{19}{-2} = -9.5 \]

Step 3: Calculate the Maximum Value

Substitute $ r = -9.5 $ back into the function to find the maximum value: \[ n(-9.5) = -(-9.5)^2 - 19(-9.5) - 25 \] \[ = -90.25 + 180.5 - 25 = 65.25 \]