Lesson 3 Rational Function (Part 1)

Summary

This video introduces rational functions by applying them to real-life scenarios, such as splitting pizza costs and calculating speed for timely school arrivals. The lesson covers the definition of rational functions, with polynomials in both the numerator and denominator. It explains how to solve rational equations using the CRAM method and interprets these functions graphically, highlighting key features like intercepts and asymptotes. The video also explores practical examples, demonstrating how the cost of sharing a pizza changes as more people join, and how to determine the horizontal and vertical asymptotes for given functions.

Takeaways

• 😀 Rational functions are fractions where both the numerator and denominator are polynomials, but the denominator cannot be zero.
• 😀 A rational function can be written as f(x) = p(x) / q(x), where p(x) and q(x) are polynomials.
• 😀 Dividing by zero in a rational function makes the function undefined, and this should be avoided.
• 😀 In real-life scenarios like sharing pizza, rational functions can model situations where something is divided among people.
• 😀 The equation 1/x = 6 / (3x + 3) can be solved using the C.R.A.M. method to find that x = 1.
• 😀 The C.R.A.M. method stands for: Clear the fractions, Remove grouping symbols, Add/Subtract like terms, and Multiply to isolate the variable.
• 😀 When graphing rational functions, as the number of people (x) increases, the shared cost per person decreases.
• 😀 The concept of asymptotes is important in rational functions; the graph approaches but never reaches a specific value.
• 😀 Rational function graphs have both x and y intercepts. The x intercepts are where f(x) = 0, and the y intercepts occur when x = 0.
• 😀 Vertical asymptotes occur where the denominator is zero, and horizontal asymptotes show the end behavior of the function.
• 😀 To find the horizontal asymptote, compare the degrees of the numerator and denominator: if they are equal, the horizontal asymptote is the ratio of their leading coefficients.

Q & A

What is a rational function?

A rational function is a fraction where both the numerator and denominator are polynomials. The denominator must not equal zero to avoid making the function undefined.

Why can the denominator of a rational function never be zero?

Dividing by zero is not allowed in mathematics because it makes the function undefined and leads to contradictions or errors.

How can we apply rational functions to real-life situations?

Rational functions can be used to model real-life scenarios such as sharing costs (e.g., splitting the cost of a pizza among a group of people) or calculating the time needed to reach a destination based on speed and distance.

What is the process to solve rational functions using the CRAM method?

The CRAM method stands for Clear the fractions, Remove grouping symbols, Add or subtract like terms, and Multiply to isolate the variable. This method helps simplify and solve rational equations step-by-step.

In the pizza example, how did the equation 1/x = 6/(3x + 3) help solve the problem?

The equation 1/x = 6/(3x + 3) represents the scenario where one pizza is shared among 'x' people, and six pizzas are shared among '3x + 3' people. Solving this equation shows that the situation holds true only when x equals 1.

What is an asymptote in the context of rational functions?

An asymptote is a boundary that a graph approaches but never actually reaches. For rational functions, vertical asymptotes occur where the denominator is zero, and horizontal asymptotes show the end behavior of the function as 'x' increases or decreases.

What does the graph of a rational function with a horizontal asymptote look like?

The graph of a rational function with a horizontal asymptote decreases or increases toward the asymptote but never actually reaches it. This represents the limiting behavior of the function as 'x' becomes very large or very small.

How do we find the x-intercepts and y-intercepts of a rational function?

To find the x-intercept, set the numerator equal to zero and solve for 'x' while ensuring the denominator is not zero. To find the y-intercept, substitute x = 0 into the function and simplify.

What are vertical and horizontal asymptotes, and how are they determined?

Vertical asymptotes occur when the denominator of a rational function equals zero, making the function undefined. Horizontal asymptotes describe the behavior of the graph as 'x' becomes very large or small, depending on the degrees of the numerator and denominator.

How do we calculate the horizontal asymptote for the function f(x) = (x^2 - 4)/(x^2 - 1)?

Since both the numerator and denominator have the same highest degree (2), the horizontal asymptote is determined by the ratio of the leading coefficients. Here, both are 1, so the horizontal asymptote is y = 1.