EN Introduction to Fractions by Slidesgo.pptx
- 2. Rational function 01 Vertical asymptotes of rational functions. 02 Horizontal asymptotes of rational functions 03 Sketch rational functions 04 Table of contents
- 4. A rational function is a function of the form: r(x)=P(x)/Q(x)
- 5. What is the slant asymptote or oblique asymptote? 02
- 6. What is the horizontal asymotote ? y = b where the graph approaches the line as the inputs approach or – ∞ ∞. A slant asymptote of a graph is a slanted line y = mx + b
- 7. Examples 1. Y = +4 2. y= Y=4 Y= 3. +3 4. Y = Y=2+3=5 Y=0
- 8. occurs when the degree of the numerator polynomial is greater than the degree of the denominator polynomial by only one degree slant asymptote or oblique asymptote
- 9. Examples y X+2 is the slant asymptote
- 10. y Y=
- 12. The Vertical Asymptotes The line x = a is a vertical asymptote of the function y = f(x) If y approaches 士 ∞ as x approaches a from the right or left
- 13. Graph of f(x) = 1/x Domain: (-∞,0) U (0,∞) Range: (-∞,0) U (0,∞) No intercepts Odd function Origin symmetry Vertical asymptote: y-axis Horizontal asymptote: x-axis
- 19. f(x) = -1 / x - 3 f(x) = 3x - 6 / x^2 -7x +10 f(x) = 8 / x^2 + 4
- 21. For the rational parent function f(x)= , • Perform vertical and horizontal shifts • Perform vertical stretches and compressions • Perform reflections across the x-axis • Perform reflections across the y-axis • Determine the transformations performed on the parent function f(x)= to get the rational function f(x)= )+k Transformations of the Rational Function
- 22. • We can represent a vertical shift of the graph of the function f(x)= by adding a constant, k, to the function: • If we shift the graph of the rational function f(x)= up 5 units, all of the points on the graph increase their y-coordinates by 5, but their x-coordinates remain the same. • Therefore, the equation of the function f(x)= +5 after it has been shifted up 5 units transforms to f(x)= +5 Vertical Shifts • If k>0, the graph shifts upwards and if k<0 the graph shifts downwards. f(x)= +k
- 23. Graph the function f(x)= - 6. What is the horizontal asymptote of the function? What is the relationship between the value of k and the horizontal asymptote? What is the vertical asymptote of the function? Solution: The horizontal asymptote is the line y= 6 − . The value of k is 6 − so the horizontal asymptote mimics the value of k. The vertical asymptote is the line x=0.
- 24. Horizontal Shifts • If we shift the graph of the function f(x)= right 7 units, all of the points on the graph increase their x-coordinates by 7, but their y- coordinates remain the same. The point (1, 1) in the original graph is moved to (8, 1). Any point (x,y) on the original graph is moved to (x+7,y) • We can represent a horizontal shift of the graph of the function f(x)= by subtracting a constant, ℎ, from the variable x. f(x)= • If h>0the graph shifts toward the right and if h<0 the graph shifts to the left.
- 25. But what happens to the original function f(x)= f(x)= ? An automatic assumption may be that since x moves to x+7that the function will become f(x)= f(x)= . But that is NOT the case. Remember that the x- intercept is moved to (8, 1) and if we substitute x=8 into the function f(x)= f(x)= we get f(x)= f(x)= ≠1!! The way to get a function value of 1 is for the transformed function to be f(x)= . Then f(8)= = 1. So the function f(x)= transforms to f(x)= after being shifted 7 units to the right. The reason is that when we move the function 7 units to the right, the x-value increases by 7 and to keep the corresponding y- coordinate the same in the transformed function, the x-coordinate of the transformed function needs to subtract 7 to get back to the original x that is associated with the original y-value.
- 26. Graph the function f(x)= . What is the vertical asymptote of the function? What is the relationship between the value of ℎ and the vertical asymptote? What is the horizontal asymptote of the function? Solution: The vertical asymptote is the line x= 6 − . The value of ℎ is −6 so the vertical asymptote mimics the value of ℎ. The horizontal asymptote is the line y=0.
- 27. Vertical and Horizontal Shifts
- 31. Reflections across the x-axis
- 32. Reflections across the y-axis y
- 33. SKETCHING GRAPHS OF RATIONAL FUNCTIONS 1. Factor: Factor the numerator and denominator. 2.Intercepts: -Find the x-intercepts by determining the zeros of the numerator and the y-intercept from the value of the function at x = 0. 3.Vertical Asymptotes: Find the vertical asymptotes by determining the zeros of the denominator, and then see whether y-> ∞ or y-› -∞o on each side of each vertical asymptote by using test values. 4.Horizontal Asymptote: Find the horizontal asymptote (if any). 5.Sketch the Graph. Graph the information provided by the first four steps. Then plot as many additional points as needed to fill in the rest of the graph of the function.
- 36. Exercises
- 37. Q1:
- 38. Q2:
- 39. Q3:
- 40. Q4:
- 41. Q5:
- 42. Q6
- 43. Q7
- 44. Q8
- 45. Q9
- 46. Q10
- 47. Q11
- 48. Sources ● 7.2: Transformations of the Rational Function | Intermedia te Algebra (lumenlearning.com) ● Ghaith sessions ● Chess worksheets
- 49. CREDITS: This template has been created by Slidesgo, and includes icons by Flaticon, infographics & images by Freepik and content by Eliana Delacour Thanks Do you have any questions?