Problem 20 - College Algebra with Corequisite Support

Question

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. $$ f(x)=-\frac{1}{3} x^{2}-2 x+3 $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Quadratic Function Form

The given quadratic function is \( f(x) = -\frac{1}{3}x^2 - 2x + 3 \). This can be written in the standard form \( ax^2 + bx + c \), where \( a = -\frac{1}{3} \), \( b = -2 \), and \( c = 3 \).

Step 2: Determine the Concavity

The concavity of the parabola is determined by the sign of \( a \). Since \( a = -\frac{1}{3} \) is negative, the parabola opens downward, meaning the function has a maximum value.

Step 3: Find the Axis of Symmetry

The axis of symmetry for a quadratic function \( ax^2 + bx + c \) is given by the formula \( x = -\frac{b}{2a} \). Substituting \( b = -2 \) and \( a = -\frac{1}{3} \), 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, substitute \( x = 3 \) into the function: \[ f(3) = -\frac{1}{3}(3)^2 - 2(3) + 3 \]Simplify the expression: \[ f(3) = -\frac{1}{3}(9) - 6 + 3 \]\[ f(3) = -3 - 6 + 3 = -6 \] The maximum value of the function is \( -6 \).

Key Concepts

Maximum Value

The maximum value of a quadratic function is a concept where we determine the highest point the function can reach on its graph. Since this quadratic equation is of the form \[ ax^2 + bx + c \] with \( a = -\frac{1}{3} \), it is open downwards, indicating a 'hill-like' shape for the parabola. Thus, it will have a maximum value.

To find this maximum value mathematically, we need to substitute the x-coordinate of the parabola's vertex, or the 'axis of symmetry,' back into the original function. For the given quadratic, the vertex occurs at \( x = 3 \). By substituting into \( f(x) = -\frac{1}{3}x^2 - 2x + 3 \), we perform:

• Calculate \( f(3) \):
• \( f(3) = -\frac{1}{3} \times 9 - 6 + 3 \)
• Simplify: \( f(3) = -3 - 6 + 3 \)
• This evaluates to \( f(3) = -6 \)

Thus, the maximum value of this quadratic function is \(-6\). This point is the peak of the parabola and indicates the highest point reached on the graph.

Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex of a parabola, effectively dividing it into two mirror-image halves. This axis of symmetry is an important feature for understanding the geometric properties of a quadratic function.

For any quadratic function expressed in the standard form \[ ax^2 + bx + c \], the formula to calculate the axis of symmetry is:

• \( x = -\frac{b}{2a} \)
• Substitute \( a = -\frac{1}{3} \) and \( b = -2 \).

After substituting, we proceed as follows:

• \( x = -\frac{-2}{2 \times (-\frac{1}{3})} \)
• Simplify to find \( x = 3 \).

The axis of symmetry, \( x = 3 \), tells us that the parabola is symmetric about this vertical line. It also helps locate the vertex, which is essential when determining the maximum or minimum value of the function.

Parabola Concavity

Parabola concavity refers to the direction in which a parabola opens. It is determined by the sign of the quadratic term's coefficient in the function's equation. Knowing the direction helps to quickly ascertain whether the function has a maximum or minimum point.

In the equation \[ f(x) = -\frac{1}{3}x^2 - 2x + 3 \], we see that

• The quadratic coefficient \( a = -\frac{1}{3} \) is less than zero.

A negative \( a \) value means the parabola open downswards, displaying a concavity like an upside-down "U" shape.

The downward opening indicates that the quadratic function reaches a peak or maximum value (and no minimum) at its vertex. Understanding the concavity gives insight into the nature of the solutions and is essential while graphing the function or predicting its behavior over a range of x-values.