Problem 97 - Applied Calculus

Question

We have discussed quadratic functions that open \(u p\) or open down. Can a quadratic function open sideways? Explain.

Step-by-Step Solution

Answer

No, a quadratic function cannot open sideways and still be a function.

Step 1: Understand the Basic Form of a Quadratic Function

A quadratic function is usually in the form of \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. The graph of this function is a parabola that opens either up or down.

Step 2: Analyze the Parabola Direction

The direction of the parabola (up or down) is determined by the sign of \( a \). If \( a > 0 \), the parabola opens upwards. If \( a < 0 \), the parabola opens downwards.

Step 3: Consider Sideways Opening

To open sideways, a quadratic function would need to have a relationship expressed in terms of \( y \), not \( x \). This would look like \( x = ay^2 + by + c \), which is not the standard form of a function as it does not express \( y \) as a single equation in terms of \( x \).

Step 4: Conclusion Based on Function Definition

By definition, a function cannot have more than one output for a single input. Hence, a quadratic function like \( x = ay^2 + by + c \) is not a function. Therefore, a quadratic function that opens sideways does not exist in a functional context.

Key Concepts

Parabola

A parabola is a specific symmetrical curve that represents the graph of a quadratic function. Imagine a "U" shape or an upside-down "U".
When you graph a quadratic function, you get this distinct shape due to the squared term (\(x^2\) or \(y^2\). Parabolas are defined by specific properties:

• The vertex, which is the highest or lowest point.
• The axis of symmetry, which is a vertical line that splits the parabola into two mirror-like halves.
• The focus and directrix, which are points that determine the parabola's width and curvature.

This shape can either open up or down depending on the function's equation. Importantly, the basic quadratic function, \( f(x) = ax^2 + bx + c \), always results in a parabola that opens vertically.

Function Definition

A function is a rule that assigns exactly one output to each input. For something to qualify as a function, every \( x \) value must have one and only one corresponding \( y \) value.
Consider the quadratic function \( f(x) = ax^2 + bx + c \). Here, every input \( x \) results in a single output \( f(x) \), which maintains the function nature.
Now, if we were to attempt a sideways-opening quadratic, like \( x = ay^2 + by + c \), we'd encounter problems. For a single input \( y \), there could be more than one \( x \) value, violating the fundamental rule of functions. Consequently, a sideways parabola doesn't work in the functional sense, as it cannot meet the function's definition.

Function Direction

The direction a parabola faces (up or down) depends on the coefficient \( a \) in the quadratic function. It's as simple as observing whether \( a \) is positive or negative.
For the function \( f(x) = ax^2 + bx + c \):

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

This characteristic is crucial because it impacts the vertex's role: when opened upwards, the vertex lies at the minimum point. When opened downwards, it’s at the maximum point.
The directionality of the parabola underlines why quadratic functions can't authentically open sideways within their standard function form.