Question

Is it possible to write two rational functions whose product when graphed is a parabola and whose quotient when graphed is a hyperbola? Justify your answer.

Short answer

It is possible to create two rational functions whose product when graphed is a parabola. However, it is impossible to have two rational functions whose quotient forms a hyperbola.

Step-by-step solution

Understanding of Rational Functions

A rational function is the division of two polynomial functions. Hence, a rational function \( f(x) \) can be expressed as \( f(x) = \frac {p(x)}{q(x)} \) where \( p(x) \) and \( q(x) \) are polynomial functions.

Product of two rational functions

The product of two rational functions is a rational function since the multiplication of any two polynomials is another polynomial. If we choose two rational functions carefully, their product can be a parabola i.e. a second degree polynomial equation. Example: \( f(x) = \frac {x}{1} \) and \( g(x) = \frac {x}{1} \). The product \( h(x) = f(x) \times g(x) = x^2 \) is a parabola in graphical representation.

Quotient of two rational functions

The quotient of two rational functions is also a rational function. However, the quotient of two rational functions may not always result in a hyperbola. A hyperbola is represented by the equation: \( \frac {(x - h)^2}{a^2} - \frac {(y - k)^2}{b^2} = 1 \), or its transposed variant, neither of which is a rational function. As a result, two rational functions' quotient will not yield a hyperbola graph.

Key concepts

Polynomial Functions

Polynomial functions are the foundation of many mathematical models and equations used in algebra. At its core, a polynomial function is an expression composed of variables and coefficients, combined using only addition, subtraction, multiplication, and non-negative integer exponents of variables. These can look like:

• Linear polynomials: such as \( f(x) = 3x + 2 \)
• Quadratic polynomials: like \( f(x) = 2x^2 + 4x + 1 \)
• Cubic and higher-degree polynomials.

Polynomials are key to understanding more complex functions, because their operations—like addition, multiplication, and division—become building blocks in rational and other types of functions. Understanding their behavior, such as how they graph on the coordinate plane, is essential for navigating more advanced topics.

Product of Rational Functions

Rational functions are essentially fractions where both the numerator and denominator are polynomials. The product of two rational functions is determined by multiplying these polynomials:- Consider two rational functions: \( f(x) = \frac{p(x)}{q(x)} \) and \( g(x) = \frac{r(x)}{s(x)} \).- Their product will be \( f(x) \times g(x) = \frac{p(x) \cdot r(x)}{q(x) \cdot s(x)} \).When examining the product of these functions, the resulting expression is a new rational function.
A fascinating real-world example is when the product of these rational functions results in a perfect square or a specific degree of polynomial, such as a parabola. In our case, if each rational function represented a linear term \( \frac{x}{1} \), their product, \( x \times x = x^2 \), becomes a quadratic polynomial which takes the shape of a parabola when graphed. This demonstrates how careful selection of functions can tailor the resulting graph to fit specific criteria.

Quotient of Rational Functions

The quotient of rational functions is similarly essential to understand. It's found by dividing one rational function by another:

• If \( f(x) = \frac{p(x)}{q(x)} \)
• and \( g(x) = \frac{r(x)}{s(x)} \),

the quotient becomes \( \frac{f(x)}{g(x)} = \frac{p(x) \cdot s(x)}{q(x) \cdot r(x)} \).This operation shows us how polynomials interact through division, building upon their already complex nature.
While graphing, rational function quotients might create various shapes, resulting generally in a rational expression rather than a geometric curve like a hyperbola. The symbolic form of a hyperbola doesn't fit within standard rational functions, as it involves subtracting squared terms. Therefore, a quotient of rational functions does not graph directly as a hyperbola but can assume more irregular shapes around specific holes and asymptotes created by the polynomial divisions.

Graphing Parabolas

Graphing parabolas is a core topic of algebra, especially when dealing with polynomial equations. A parabola is the graph of a quadratic function, like \( y = ax^2 + bx + c \). It forms a distinct U-shape that can open either upwards or downwards based on the coefficient of the squared term.Key elements to notice when graphing include:

• The vertex, which is the peak or lowest point.
• The axis of symmetry, which is a vertical line that splits the parabola into mirror images.
• The direction it opens (up or down), determined by the sign of \( a \).

Being familiar with how parabolas behave helps students connect the dots between algebraic equations and their graphical interpretations. When two rational functions, like those illustrated before, are multiplied to form \( x^2 \), graphing this result will provide the visual of a clear, upward-opening parabola. Understanding this transformation bridges the abstract with the observable.

Graphing Hyperbolas

Hyperbolas are fascinating curves characterized by two distinct branches. Think of them as opposite twins that mirror each other. Unlike parabolas, hyperbolas don't represent rational functions directly. Instead, they take on a unique form:The standard hyperbola equation is \( \frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1 \) or its rotated counterpart. Key features include:

• Two branches, opening opposite directions.
• An asymptotic structure, approaching but never touching specific lines.
• A center point denoted by \( (h, k) \).

In the context of rational functions, creating a graph that resembles a hyperbola through quotients isn't straightforward due to the non-linear, non-rational terms like subtraction of squares involved in their equations. While rational function quotients can produce graphs with complex behavior, modeling an exact hyperbola requires specific algebraic manipulation outside the bounds of basic rational expressions.