Problem 98 - Brief Applied Calculus

Question

For the quadratic function \(f(x)=a x^{2}+b x+c\), what condition on one of the coefficients will guarantee that the function has a highest value? A lowest value?

Step-by-Step Solution

Answer

Highest value: \(a < 0\). Lowest value: \(a > 0\).

Step 1: Understand the Shape of Quadratic Functions

A quadratic function follows the form \(f(x) = ax^2 + bx + c\). It represents a parabola, which opens upwards if \(a > 0\) and opens downwards if \(a < 0\).

Step 2: Identify Condition for Highest or Lowest Value

The coefficient \(a\) determines whether the parabola opens upward or downward. For a highest value, the parabola must open downward (\(a < 0\)), and for a lowest value, it must open upward (\(a > 0\)).

Step 3: Conclusion on Conditions

The condition for the quadratic function \(f(x) = ax^2 + bx + c\) to have a highest value is \(a < 0\), while the condition for it to have a lowest value is \(a > 0\).

Key Concepts

Understanding the Parabola

In mathematics, a parabola is a symmetrical curve that is formed as the graph of a quadratic function. When you see a quadratic function like \( f(x) = ax^2 + bx + c \), it will graph a shape known as a parabola. Parabolas have unique properties which depend on the value of the coefficient \( a \). They can either open upwards or downwards, determining whether they have a highest or a lowest point.

• If \( a > 0 \), the parabola opens upwards like a smiley face ☺️.
• If \( a < 0 \), the parabola opens downwards like a frown ☹️.

This orientation is crucial because it helps us understand whether the parabola will have a peak or a valley. The point at which the parabola changes direction is called the "vertex," and it represents the maximum or minimum value of the function.

The Role of Coefficient "a"

The coefficient "a" in the quadratic function \( f(x) = ax^2 + bx + c \) is essential in shaping the parabola. Let's break down what this coefficient does:

• Controls the opening direction: As mentioned, if \( a > 0 \), the parabola opens upwards, and if \( a < 0 \), it opens downwards. This opening direction is vital for determining the function's extreme values—highest or lowest.
• Influences the "width" of the parabola: The absolute value of \( a \) controls how "wide" or "narrow" the parabola appears. A larger absolute value results in a narrower parabola, while a smaller absolute value results in a wider one.

By understanding how "a" affects the parabola, we gain insight into both the algebraic and graphical behaviors of quadratic functions. Also, note that when \( a = 0 \), it ceases to be a quadratic function. Instead, it becomes linear, creating a straight line rather than a curved parabola.

Finding the Highest and Lowest Values

The highest or lowest values of a quadratic function occur at what we call the "vertex" of the parabola. To determine if a quadratic function has a highest or lowest value, you need to examine the sign of the coefficient "a":

• For a highest value, as found in a "downward-opening" parabola, the condition is \( a < 0 \). This alignment implies that the vertex of the parabola is at its peak, representing the highest point of the curve.
• For a lowest value, associated with an "upward-opening" parabola, the condition is \( a > 0 \). Here, the vertex is at its lowest point, standing as the bottom-most point of the curve.

Understanding these conditions helps in solving real-world problems and optimizing solutions, such as determining maximum profits or minimum costs. By examining the sign of "a," you can quickly decide whether you are dealing with a function that has a peak or a valley, allowing you to predict and interpret its behavior accurately.