Representations of Rational Function

Cris Cruz
18 Oct 202216:10

Summary

TLDRThis lesson focuses on rational functions, teaching how to represent them through equations, tables of values, and graphs. It introduces the simplest form, f(x) = 1/x, and demonstrates creating a table with various x values, including handling undefined cases. The script then illustrates graphing the function, highlighting the concept of asymptotes. It further applies this knowledge to model real-life situations, such as a runner's speed over time, and concludes with a bonus discussion on transforming graphs of rational functions, including shifts and reflections, enhancing understanding of their behavior.

Takeaways

  • 📚 The lesson's main objective is to understand how to represent a rational function through its equation, table of values, and graph.
  • 🔍 A rational function is defined as a function of the form f(x) = P(x) / Q(x), where P(x) and Q(x) are polynomials and Q(x) ≠ 0.
  • 📈 The simplest rational function example given is f(x) = 1/x, which requires considering values for x that are negative, zero, and positive.
  • 📊 To create a table for a rational function, substitute values of the independent variable into the function's equation and calculate the corresponding dependent variable values.
  • 📉 The graph of a rational function will have vertical asymptotes where the denominator is zero, and these are represented by dashed lines on the graph.
  • 🏃‍♂️ An example of a real-life application of a rational function is modeling the speed of a runner over time, with the function S(t) = 100 / t.
  • 📝 When graphing, only positive values for time are considered since time cannot be negative.
  • 📉 The graph of the runner's speed function will show how speed decreases as time increases.
  • 🔄 Transformations of rational functions can involve horizontal shifts (changing the asymptote) and vertical shifts of the graph.
  • 🔄 Changes in the denominator of a rational function can result in vertical shifts of the graph, while changes in the numerator can result in horizontal shifts.
  • 📈 The graph of 1/x² is an example where the function does not have a negative denominator, and the graph is flipped on the x-axis compared to 1/x.
  • 📚 The lesson concludes with a summary of how to represent rational functions and an invitation to check understanding through practice.

Q & A

  • What is the main objective of the lesson discussed in the transcript?

    -The main objective of the lesson is to represent a rational function through its equation, table of values, and graph, and as a bonus, to discuss how to transform graphs of rational functions.

  • What is a rational function in mathematical terms?

    -A rational function is a function of the form f(x) = P(x) / Q(x), where P(x) and Q(x) are polynomial functions and Q(x) is not equal to zero.

  • Why is it important to include negative, zero, and positive values when creating a table for a rational function?

    -Including negative, zero, and positive values ensures that the behavior of the rational function is understood across the entire domain, except where the function is undefined (e.g., division by zero).

  • What is an asymptote and why is it significant in the graph of a rational function?

    -An asymptote is a line that a graph approaches but never intersects. It is significant in the graph of a rational function because it indicates the values that the function approaches as the independent variable approaches infinity or negative infinity.

  • How does the value of a function become undefined in the context of rational functions?

    -The value of a function becomes undefined when the denominator of the rational function is zero, as division by zero is undefined in mathematics.

  • What is the world record for the 100-meter dash as mentioned in the transcript, and who holds it?

    -The world record for the 100-meter dash, as of October 2015, is 9.58 seconds, set by the Jamaican sprinter Usain Bolt.

  • How is the speed of a runner represented as a function of time in the transcript?

    -The speed of a runner is represented as a function of time by the equation S(T) = 100 / T, where S is the speed and T is the time taken to run 100 meters.

  • What is the significance of the vertical dashed line in the graph of the function S(T)?

    -The vertical dashed line in the graph of the function S(T) represents the time at which the speed would be undefined, which in this case is when T equals zero, as time cannot be negative.

  • What happens to the horizontal asymptote when a constant is added to the numerator of a rational function?

    -When a constant is added to the numerator of a rational function, the graph of the function moves vertically by the same amount, and the horizontal asymptote also shifts by that constant value.

  • What is the effect of adding a constant to the denominator of a rational function on the graph and its vertical asymptote?

    -Adding a constant to the denominator of a rational function shifts the graph horizontally. The vertical asymptote moves to the left by the absolute value of the constant if it's positive, or to the right if it's negative.

  • How does the exponent of the variable in the denominator affect the graph of a rational function?

    -If the exponent of the variable in the denominator is even, the graph will not extend into the negative y-values, effectively flipping the lower half of the graph onto the x-axis.

  • What is the process of transforming the graph of the function f(x) = 1/x^2 when a constant is added to the function?

    -When a constant is added to the function f(x) = 1/x^2, the graph moves vertically upward by the value of the constant. For example, if the function is g(x) = 1/x^2 + 2, the graph of g will be 2 units higher than that of f.

  • How does the graph of a rational function change when a constant is subtracted from the variable in the denominator?

    -When a constant is subtracted from the variable in the denominator of a rational function, the graph moves horizontally to the right by the absolute value of the constant.

  • What is the final step in the process of representing a rational function through its equation, table of values, and graph?

    -The final step is to connect the points plotted on the graph to visualize the behavior of the rational function, including its approach to any asymptotes.

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الوسوم ذات الصلة
Rational FunctionsMathematicsGraphingEquationsTables of ValuesFunction TransformationEducational ContentAsyptotesAlgebraTrigonometrySpeed Analysis
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