Problem 16 - 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. $$f(x)=-x^{2}+4 x+3$$

Step-by-Step Solution

Answer

The function has a maximum value of 7 at an axis of symmetry \(x = 2\).

Step 1: Identify the Quadratic Function Form

The given quadratic function is \(f(x) = -x^2 + 4x + 3\). This is in the standard form \(f(x) = ax^2 + bx + c\) where \(a = -1\), \(b = 4\), and \(c = 3\).

Step 2: Determine the Direction of the Parabola

Since the coefficient \(a = -1\) is negative, the parabola opens downwards. Therefore, the function will have a maximum value, not a minimum.

Step 3: Find the Axis of Symmetry

The axis of symmetry for a quadratic function in the form \(ax^2 + bx + c\) is given by the formula \(x = -\frac{b}{2a}\). Substitute \(a = -1\) and \(b = 4\) into the formula:\[x = -\frac{4}{2(-1)} = \frac{4}{2} = 2\]Thus, the axis of symmetry is \(x = 2\).

Step 4: Calculate the Maximum Value

To find the maximum value, substitute the axis of symmetry \(x = 2\) back into the function:\[f(2) = -2^2 + 4(2) + 3\]\[f(2) = -4 + 8 + 3\]\[f(2) = 7\]Therefore, the maximum value of the function is 7.

Key Concepts

Axis of Symmetry

In quadratic functions, the axis of symmetry is an imaginary line that vertically cuts the graph of the parabola into two symmetric halves. Its main purpose is to help locate the vertex of the parabola, which is the highest or lowest point on the graph, depending on its orientation. For a quadratic equation of the form \(ax^2 + bx + c\), the axis of symmetry can be calculated using the formula \(x = -\frac{b}{2a}\).

This formula is derived from the standard quadratic formula and works because it finds the horizontal line at which the two halves of the parabola mirror each other.

• The axis of symmetry helps in solving quadratic problems as it simplifies finding the vertex.
• If you know the axis of symmetry, it becomes much easier to calculate the maximum or minimum value of the parabola.
• In our example, since \(a = -1\) and \(b = 4\), we calculated the axis of symmetry as \(x = 2\).

Vertex Form

Quadratic functions can also be expressed in vertex form, which makes it easier to identify the vertex directly. While the standard form is \(ax^2 + bx + c\), the vertex form is \(f(x) = a(x-h)^2 + k\), where \((h, k)\) is the vertex of the parabola.

Converting from standard to vertex form involves completing the square or using the axis of symmetry to rearrange the equation. This transformation is useful for:

• Quickly identifying the vertex without further calculations.
• Understanding the graph's direction and shifts more intuitively.
• Making it simpler to graph the parabola, especially by hand.

In the original exercise, once we identify our axis of symmetry, finding the vertex becomes straightforward, emphasizing the utility of the vertex form in visualization and graphing tasks.

Maximum and Minimum of Functions

Determining the maximum or minimum values of a quadratic function is a key feature of analyzing these functions. When a quadratic function is graphed, it forms a parabola, which will either open upwards or downwards based on the sign of the coefficient \(a\).

• If \(a > 0\), the parabola opens upwards, and the vertex represents the minimum value.
• If \(a < 0\), the parabola opens downwards, and the vertex is where the maximum value occurs.
• In our problem, the parabola opens downward because \(a = -1\), indicating that it has a maximum value.

To find this maximum value, you substitute the axis of symmetry back into the function. For example, using \(x = 2\) in our exercise, calculating \(f(2)\) gives us the maximum value of 7. This is crucial for practical problems where identifying extremes is necessary, such as in profit maximization or finding optimal conditions.