Problem 94 - Calculus Volume 1

Question

Draw a graph to support your explanation. Can you have a finite absolute maximum for \(y=a x^{2}+b x+c\) over \((-\infty, \infty) ?\) Explain why or why not using graphical arguments.

Step-by-Step Solution

Answer

Yes, a finite absolute maximum exists when the parabola opens downward (\( a < 0 \)).

Step 1: Understanding the Quadratic Function

The function given is a quadratic function of the form \( y = ax^2 + bx + c \). The graph of a quadratic function is a parabola, which can either open upwards or downwards depending on the sign of the coefficient \( a \).

Step 2: Determine the Direction of the Parabola

For \( y = ax^2 + bx + c \), if \( a > 0 \), the parabola opens upwards, and if \( a < 0 \), the parabola opens downwards.

Step 3: Identify Absolute Maximum from Graph Shape

If the parabola opens upwards (\( a > 0 \)), the lowest point on the graph is the vertex, meaning the function has an absolute minimum but no finite maximum over \((-fty, fty)\). Conversely, if the parabola opens downwards (\( a < 0 \)), the highest point, the vertex, becomes an absolute maximum.

Step 4: Analyze for Finite Absolute Maximum

Since we focus on the existence of a finite absolute maximum over \( (-fty, fty) \), this can only occur when \( a < 0 \) because the vertex then represents the highest point. When \( a > 0 \), the function continues to increase without bound, thereby having no finite maximum.

Step 5: Conclusion Through Graphical Representation

Graphically, when \( a > 0 \), the parabola keeps rising on both sides, indicating an infinite maximum. When \( a < 0 \), the parabola caps at the vertex, giving a finite maximum. Therefore, a finite absolute maximum only exists if \( a < 0 \).

Key Concepts

Parabola

In the realm of quadratic functions, the term "parabola" holds a special place. This is the shape of the graph described by the quadratic equation \( y = ax^2 + bx + c \). Simply put, a parabola is a symmetrical, U-shaped curve.

Depending on the value of the coefficient \( a \), the parabola can open upwards or downwards:

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

Understanding whether a parabola opens upwards or downwards is crucial for analyzing its behavior and the related concepts.

Vertex

The vertex is a pivotal point on a parabola and offers significant insights about quadratic functions.

It acts as the turning point:

• For a parabola opening upwards, it marks the lowest point or the absolute minimum.
• For a parabola opening downwards, it signals the highest point or the absolute maximum.

You can find the vertex using the formula \( x = \frac{-b}{2a} \) for its x-coordinate. The y-coordinate is determined by plugging this x-value back into the quadratic function. Recognizing the vertex helps in understanding the extreme values of the function.

Absolute Maximum

The concept of an absolute maximum revolves around identifying the highest value a function can attain over its domain.

In the context of quadratic functions, an absolute maximum exists if the parabola opens downwards.

• This occurs when the coefficient \( a < 0 \), with the vertex marking this maximum.
• When \( a > 0 \), the function continues to rise without bound, implying no end point or finite maximum in the real number domain \((-fty, fty)\).

Hence, for functions with \( a < 0 \), the vertex is paramount, as it represents the peak value or maximum of the function.

Coefficient Sign

The sign of the coefficient \( a \) in the quadratic equation is more than just a number; it dictates the graph's overall direction.

It's vital for deciding the parabola's orientation:

• A positive \( a \) results in an upwards-opening parabola, indicating a minimum value without constraints on the maximum potential rise.
• A negative \( a \) produces a downwards-opening parabola, setting a cap on its ascent with the vertex serving as the maximum.

Understanding how the coefficient sign influences the parabola is crucial for effectively analyzing and predicting the graph's behavior.

Graphical Analysis

Graphical analysis offers a tangible way to understand the behavior of a quadratic function through its visual representation.

By interpreting the parabola:

• The function's direction and maximum or minimum values become readily apparent.
• Graphs illustrate how the parabolas ascend or descend indefinitely, contingent on the sign of \( a \).
• The vertex provides a clear point of reference among the infinite paths the parabola could take.

Using a graph to analyze the function empowers you to draw meaningful conclusions about the nature of the function and recognize key features like vertex and maximums or minimums.