Questions: For the following exercises, determine whether there is a minimum or maximum value to each quadratic function. Find the value and the axis of symmetry. 19. f(x) = (1/2) x^2 + 3x + 1

For the following exercises, determine whether there is a minimum or maximum value to each quadratic function. Find the value and the axis of symmetry.
19. f(x) = (1/2) x^2 + 3x + 1

Transcript text: For the following exercises, determine whether there is a minimum or maximum value to each quadratic function. Find the value and the axis of symmetry. 19. $f(x)=\frac{1}{2} x^{2}+3 x+1$

Solution

Solution Steps

To determine whether there is a minimum or maximum value for the quadratic function $ f(x) = \frac{1}{2} x^2 + 3x + 1 $, we need to look at the coefficient of $ x^2 $. Since it is positive ($\frac{1}{2}$), the parabola opens upwards, indicating a minimum value. The axis of symmetry can be found using the formula $ x = -\frac{b}{2a} $, where $ a $ and $ b $ are the coefficients from the quadratic equation $ ax^2 + bx + c $. The minimum value can then be found by substituting this $ x $-value back into the function.

Step 1: Determine the Axis of Symmetry

The axis of symmetry for the quadratic function $ f(x) = \frac{1}{2} x^2 + 3x + 1 $ is calculated using the formula:

\[ x = -\frac{b}{2a} \]

Substituting the values $ a = \frac{1}{2} $ and $ b = 3 $:

\[ x = -\frac{3}{2 \cdot \frac{1}{2}} = -3 \]

Step 2: Find the Minimum Value

Since the coefficient of $ x^2 $ is positive, the function has a minimum value. We find this minimum value by substituting $ x = -3 $ back into the function:

\[ f(-3) = \frac{1}{2}(-3)^2 + 3(-3) + 1 \]

Calculating this gives:

\[ f(-3) = \frac{1}{2}(9) - 9 + 1 = 4.5 - 9 + 1 = -3.5 \]