Problem 19 - 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. $$f(x)=\frac{1}{2} x^{2}+3 x+1$$

Step-by-Step Solution

Answer

The function has a minimum value of -3.5 at the axis of symmetry x = -3.

Step 1: Identify a, b, and c

The quadratic function is given in the form \(f(x) = ax^2 + bx + c\). For the function \(f(x) = \frac{1}{2}x^2 + 3x + 1\), we identify the coefficients as follows: \(a = \frac{1}{2}\), \(b = 3\), and \(c = 1\).

Step 2: Determine the direction of the parabola

Since the coefficient \(a = \frac{1}{2}\) is positive, the parabola opens upwards. This means the function has a minimum value at its vertex.

Step 3: Calculate the axis of symmetry

The axis of symmetry for a quadratic function \(ax^2 + bx + c\) is given by the formula \(x = -\frac{b}{2a}\). Substitute the values of \(b = 3\) and \(a = \frac{1}{2}\):\[ x = -\frac{3}{2 \times \frac{1}{2}} = -3 \]Thus, the axis of symmetry is \(x = -3\).

Step 4: Find the vertex and minimum value

To find the vertex, substitute \(x = -3\) into the original function \(f(x) = \frac{1}{2}x^2 + 3x + 1\):\[f(-3) = \frac{1}{2}(-3)^2 + 3(-3) + 1 \= \frac{1}{2} \times 9 - 9 + 1 \= 4.5 - 9 + 1 \= -3.5\]Thus, the vertex is at \((-3, -3.5)\), and the minimum value of the function is \(-3.5\).

Key Concepts

Minimum and Maximum Values

Quadratic functions are critical in mathematics, especially when analyzing their graphs to find minimum or maximum values. These values occur at the vertex of the parabola. Determining whether a quadratic function has a minimum or maximum value depends on the direction in which the parabola opens.

• If the coefficient of the term with the highest degree, \(a\) in \(f(x) = ax^2 + bx + c\), is positive, the parabola opens upwards. In this case, the function achieves a minimum value at its vertex.
• If \(a\) is negative, the parabola opens downward, and the function reaches a maximum value at the vertex.

Understanding this will help you quickly identify whether you're seeking a minimum or maximum when analyzing the function. In the exercise, since \(a = \frac{1}{2}\) is positive, the quadratic function \(f(x) = \frac{1}{2}x^2 + 3x + 1\) has a minimum value at its vertex.

Parabola

A parabola is the graph of a quadratic function and is always shaped like a U, either opening upwards or downwards.

• The openness is determined by the sign of the coefficient \(a\).
• If \(a > 0\), the parabola opens upwards. If \(a < 0\), it opens downwards.
• Parabolas are symmetrical, meaning they mirror evenly about a central line.

The depiction of a parabola helps visualize where the minimum or maximum value occurs and makes it easier to understand the nature of the function's solutions. For instance, in this exercise, since the parabola opens upwards, our focus was on finding its minimum value.

Axis of Symmetry

The axis of symmetry is an essential feature of a parabola, ensuring it is symmetrical. It is a vertical line that splits the parabola into two mirror-image halves.

• The formula to find the axis of symmetry in a quadratic equation \(f(x) = ax^2 + bx + c\) is \(x = -\frac{b}{2a}\).
• Finding this axis is a crucial step because it is where the vertex lies, and thus, where the minimum or maximum value is located.

In our problem, we calculated the axis of symmetry as \(x = -3\), which serves as the midpoint for the parabola, helping us pinpoint the location of the vertex and the minimum value precisely.

Vertex

The vertex plays a pivotal role in understanding quadratic functions as it represents the point where the parabola achieves its minimum or maximum value.

• Once the axis of symmetry is known, substitute its value into the original function to find the y-coordinate of the vertex.
• The vertex is expressed as \((x, y)\), where \(x\) is the axis of symmetry and \(y\) is the function value at this point.
• In any quadratic equation, the vertex provides significant insight into the behavior and graph of the function.

For our exercise, substituting \(x = -3\) into the quadratic equation, we arrive at the vertex \((-3, -3.5)\). This indicates the function's minimum value is \(-3.5\), which occurs at \(x = -3\). Grasping the concept of the vertex is crucial for effectively tackling problems involving quadratic functions.