Questions: Polynomial and Rational Functions Finding the maximum or minimum of a quadratic function Answer the questions below about the quadratic function. f(x)=3x^2+30x+74 Does the function have a minimum or maximum value? Minimum Maximum What is the function's minimum or maximum value? Where does the minimum or maximum value occur? x=

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

Answer the questions below about the quadratic function.
f(x)=3x^2+30x+74

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

What is the function's minimum or maximum value?

Where does the minimum or maximum value occur?
x=

Transcript text: Polynomial and Rational Functions Finding the maximum or minimum of a quadratic function Answer the questions below about the quadratic function. \[ f(x)=3 x^{2}+30 x+74 \] Does the function have a minimum or maximum value? Minimum Maximum What is the function's minimum or maximum value? $\square$ Where does the minimum or maximum value occur? \[ x= \] $\square$

Solution

Solution Steps

To determine whether the quadratic function $ f(x) = 3x^2 + 30x + 74 $ has a minimum or maximum value, we need to look at the coefficient of the $ x^2 $ term. Since the coefficient is positive (3), the parabola opens upwards, indicating a minimum value.

To find the minimum value and the point at which it occurs, we use the vertex formula for a quadratic function $ ax^2 + bx + c $. The x-coordinate of the vertex is given by $ x = -\frac{b}{2a} $. Once we have the x-coordinate, we can substitute it back into the function to find the minimum value.

Step 1: Determine if the function has a minimum or maximum value

Given the quadratic function $ f(x) = 3x^2 + 30x + 74 $, we observe that the coefficient of $ x^2 $ is positive ($ a = 3 $). Therefore, the parabola opens upwards, indicating that the function has a minimum value.

Step 2: Find the x-coordinate of the vertex

The x-coordinate of the vertex of a quadratic function $ ax^2 + bx + c $ is given by: \[ x = -\frac{b}{2a} \] Substituting $ a = 3 $ and $ b = 30 $: \[ x = -\frac{30}{2 \cdot 3} = -\frac{30}{6} = -5 \]

Step 3: Calculate the minimum value of the function

To find the minimum value, substitute $ x = -5 $ back into the function $ f(x) $: \[ f(-5) = 3(-5)^2 + 30(-5) + 74 \] \[ f(-5) = 3 \cdot 25 - 150 + 74 \] \[ f(-5) = 75 - 150 + 74 = -1 \]