Questions: Polynomial and Rational Functions Finding the maximum or minimum of a quadratic function Answer the questions below about the quadratic function. f(x)=-2x^2-12x-20 Does the function have a minimum or maximum value? Minimum Maximum Where does the minimum or maximum value occur? x= What is the function's minimum or maximum value?

Polynomial and Rational Functions
Finding the maximum or minimum of a quadratic function

Answer the questions below about the quadratic function.
f(x)=-2x^2-12x-20

Does the function have a minimum or maximum value?
Minimum Maximum

Where does the minimum or maximum value occur?
x=

What is the function's minimum or maximum value?

Transcript text: Polynomial and Rational Functions Finding the maximum or minimum of a quadratic function Answer the questions below about the quadratic function. \[ f(x)=-2 x^{2}-12 x-20 \] Does the function have a minimum or maximum value? Minimum Maximum Where does the minimum or maximum value occur? \[ x= \] $\square$ What is the function's minimum or maximum value? $\square$

Solution

Solution Steps

To determine whether the quadratic function has a minimum or maximum value, we need to look at the coefficient of the $x^2$ term. If it is positive, the parabola opens upwards and has a minimum; if it is negative, the parabola opens downwards and has a maximum. For the given function, the coefficient is negative, indicating a maximum value. The vertex of the parabola gives the x-coordinate where this maximum occurs, which can be found using the formula $x = -\frac{b}{2a}$. Finally, substitute this x-coordinate back into the function to find the maximum value.

Step 1: Determine Maximum or Minimum Value

The given quadratic function is $ f(x) = -2x^2 - 12x - 20 $. Since the coefficient of $ x^2 $ is negative ($ a = -2 $), the parabola opens downwards, indicating that the function has a maximum value.

Step 2: Find the x-coordinate of the Vertex

The x-coordinate of the vertex, where the maximum occurs, is calculated using the formula: \[ x = -\frac{b}{2a} \] Substituting the values $ b = -12 $ and $ a = -2 $: \[ x = -\frac{-12}{2 \cdot -2} = -3.0 \]

Step 3: Calculate the Maximum Value

To find the maximum value of the function, substitute $ x = -3.0 $ back into the function: \[ f(-3.0) = -2(-3.0)^2 - 12(-3.0) - 20 = -2(9) + 36 - 20 = -18 + 36 - 20 = -2.0 \]