Problem 20 - College Algebra

Question

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

Step-by-Step Solution

Answer

The maximum value is -6, and the axis of symmetry is x = 3.

Step 1: Identify the Coefficients

First, we need to identify the coefficients of the quadratic function in the form \( f(x) = ax^2 + bx + c \). In this case, the function is \( f(x) = -\frac{1}{3}x^2 - 2x + 3 \). Thus, \( a = -\frac{1}{3} \), \( b = -2 \), and \( c = 3 \).

Step 2: Determine the Direction of the Parabola

A quadratic function \( f(x) = ax^2 + bx + c \) opens downward if \( a < 0 \) and upward if \( a > 0 \). In this case, \( a = -\frac{1}{3} < 0 \), so the parabola opens downward, meaning there is a maximum point.

Step 3: Find the Axis of Symmetry

The axis of symmetry for a quadratic function \( f(x) = ax^2 + bx + c \) is given by the formula \( x = -\frac{b}{2a} \). Substituting the coefficients, we have \( x = -\frac{-2}{2(-\frac{1}{3})} = -\frac{-2}{-\frac{2}{3}} = 3 \). Thus, the axis of symmetry is \( x = 3 \).

Step 4: Calculate the Maximum Value

To find the maximum value of the function, substitute the axis of symmetry \( x = 3 \) back into the function: \[ f(3) = -\frac{1}{3}(3)^2 - 2(3) + 3 \]. This simplifies to \[ f(3) = -\frac{1}{3}(9) - 6 + 3 = -3 - 6 + 3 = -6 \]. So, the maximum value is \(-6\).

Key Concepts

Understanding the Parabola

When working with quadratic functions, a key feature is the parabola. A parabola is the U-shaped curve that graphically represents the equation of a quadratic function. It can open either upwards or downwards.
- If the parabola opens upwards, it looks like a U shape with the ends pointing upwards. In this case, it has a minimum point.- If the parabola opens downwards, it’s like an upside-down U, which means it has a maximum point.

The direction in which the parabola opens depends on the coefficient 'a' in the quadratic equation of the form \( ax^2 + bx + c \). Here's how:

• If \( a > 0 \), the parabola opens upwards and has a minimum point.
• If \( a < 0 \), the parabola opens downwards, like in our example \( f(x) = -\frac{1}{3}x^2 - 2x + 3 \), thus having a maximum point.

The Axis of Symmetry

The axis of symmetry is a crucial concept in understanding quadratics. It is the vertical line that divides the parabola into two mirror-image halves.
In the equation \( f(x) = ax^2 + bx + c \), the axis of symmetry can be calculated using the formula:

• \( x = -\frac{b}{2a} \)

In our example, we identified \( b = -2 \) and \( a = -\frac{1}{3} \). Plugging these into the formula, we find:

• \( x = -\frac{-2}{2(-\frac{1}{3})} = 3 \)

This tells us that the line \( x = 3 \) is the axis of symmetry. This line is exactly where the parabola can be folded over itself evenly, showcasing the balance of the quadratic curve.

Finding the Maximum Value

In the context of quadratics, the maximum value is the highest point on the parabola when it opens downwards. For a parabola that opens upwards, we refer instead to a minimum value.
To find this value for our function \( f(x) = -\frac{1}{3}x^2 - 2x + 3 \), we substitute the x-value of the axis of symmetry back into the function.

• Substitute \( x = 3 \) into the function: \( f(3) = -\frac{1}{3}(3)^2 - 2(3) + 3 \)
• Simplify the equation: \( f(3) = -\frac{1}{3}(9) - 6 + 3 = -3 - 6 + 3 = -6 \)

Thus, the maximum value of the parabola in this quadratic function is \(-6\). It’s important to note that this value represents the peak of the downward-opening parabola, indicating the highest point the curve reaches.