8 Rational Equations Algebra Review Express each sum as a single fraction by finding the least common denominator. 1. 3 I 2. 3. 1 3 I+1 - +5 2 4 z - 9 4. I+3 x2 + 8x + 15 22Practice: Identifying Features of Rational Functions Using the information from the Rational Functions activity, find (a) the domain of each function in interval notation and b) all asymptotes (vertical, horizontal, and/or slant) and holes for the graph of each function. You may need to use long division to find the slant asymptotes. 1. h(x) = 4x2 + 4x + 3 x- 1 2. f(I) = 2x - 3 3. s( t ) = to - 13 - t 4+4 + 5\f10 Rational Graphing and Inequalities First, recall an important fact about polynomials. 1. Suppose r = r is a zero of a polynomial p. Fill in the blanks. . If x = r has . multiplicity, then the graph of y = p(x) crosses the r-axis at I = r. . If x = r has - multiplicity, then the graph of y = p(x) touches but does not cross the r-axis at I = r. 2. We can rephrase the fact in #1 in the following way. . If x = r has odd multiplicity, then the value of y = p(I) its sign to either side of C=r. changes / does not change . If x = r has even multiplicity, then the value of y = p(x) its sign to either side changes / does not change of x = r. Model 1: Multiplicities and Vertical Asymptotes Open the Desmos activity found at this link. The link is also provided on Canvas. You may choose "Continue without signing in." 3. Fill in sign charts for each of the functions graphed on Slide 1. Label each interval of the r-axis with a "+" or a "_" corresponding with the sign of the y-values on the graph. y1 : y2 : O- -1 2 -1 3 y3 : y4 : O- 3 4. Note the connection between the sign patterns and the formulas, then move to Slide 2 and write a formula to match the graph. Check your answer by graphing it. y = 5. Use what you learned in Model 1 to fill in the blanks. Suppose f(x) = 2 is a fully simplified rational function, and x = r is a zero of q. Then the line c = r is a vertical asymptote of the graph of y = f(x), and . if x = r has odd multiplicity, then the value of y = f(x) - its sign to either side of changes / does not change c=r. . if x = r has even multiplicity, then the value of y = f(x) its sign to either side changes / does not change of T = r.Model 2: Graphing a Rational Function Our goal is to graph the function f (x) = (x -2)(2 - 7x + 10) x2 - 8x + 15 In the following problems, draw the features of the graph on the axes below as you find them. -....... .. ... 5 ... ... . .. .. . 4 . ... . .... ..;. ..... .. ... . . .. .. .....". Not. . ....... .} . . . ... . . . ..?"' . ...... ... Co - -5 -4 -3 - 2 . CO . .... -1 .. .. ...j.. -2. . .... ..... .... .. ....... -3 - ...... 6. Rewrite the function with its numerator and denominator fully factored, then write the domain of f as a union of intervals. 7. Write a simplified formula for f(r), then write the equations of any vertical asymptotes of its graph. Open Slide 1 of this Desmos activity to check your answer. 8. Find all r- and y-intercepts of the graph. Use Slide 2 to check your answer.9. There is another a value excluded from the domain (different from the vertical asymptote). Plug this value into the simplified formula for f(x), and you will get a point that the graph passes through but is excluded. Indicate this point with an open circle. Use Slide 3 to check your answer. 10. Determine whether the graph has a horizontal asymptote, slant asymptote, or neither. If it has one, find its equation. Use Slide 4 to check your answer. 11. Complete the following statement. Use Slide 5 to check your answer. Since the r-intercept came from a zero with even / odd - multiplicity, the y-coordinate of the graph sign to either side of the intercept. changes / does not change 12. Complete the following statement. Use Slide 6 to check your answer. Since the vertical asymptote came from a zero with even / odd - multiplicity, the y-coordinate of the graph sign to either side of the intercept. changes / does not change 13. Finally, connect everything you have so far with a smooth curve. Make sure that the curve tends toward the slant asymptote as the x values get larger. Use Slide 7 to check your answer.Model 1: Two Cyclists Two cyclists, Arlene and Bianca, complete a 60-mile course. Bianca's speed was 5 mph faster than Arlene's, and she finished the course 1 hour sooner. What was each cyclist's speed? Distance (mi) Speed (mph) Time (hr) Arlene Bianca 5. Fill in the missing values in the Distance column. 6. Fill in expressions for Bianca's speed and time, using s and t to refer to Arlene's speed and time. 7. Set up the equation distance = time for Arlene's data and Bianca's data. You should have two equations. speed 8. One of your equations should have t by itself (revise #3 if not). Substitute this expression for t into the other equation. 9. Solve the equation in #4. Start by writing every term with a common denominator. 10. Write your final answer to the question in Model 1 as a complete sentence. 21Model 2: Excluded Values Consider the following equation. 1 2 I - 2 x - 4 x2 - 61 +8 11. Rewrite the equation with all denominators fully factored. 12. List all x-values that would cause division by zero in any of the fractions. These are known as excluded values. 13. Solve the equation. Start by writing every term with a common denominator. 14. Check your solutions in #9 against the excluded values in #8. Discard any excluded values from your solution set.Practice: Solving Radical and Rational Equations Solve the equation. Remember to check for extraneous solutions if applicable. 1. 0 = V20 -8 -4 2. y- 1 = V-20 + 8y 3. (1 - 7)5 =2 4. (t - 5)3 =-5 275. 2 12 - 10x + 24 3 x - 6 I - 4 6. W - 7 w - 4 w - 5 -1= - w - 7 7. 9 -12 x - 8 x - 2 x2 - 10x + 16 289 Rational Functions Model 1: Building a Rectangle orianna is challenged to build a rectangle that has an area of 10 square meters. She considers different possibilities for the rectangle's dimensions as shown in the table below. Length (m) 10 5 2 1 0.5 0.1 0.05 0.01 0.005 0.001 Width (m) 1 2 1. Fill in the missing Width values in Brianna's table. 2. As the Length values get smaller, what happens to the corresponding Width values? Choose the best answer. You may want to consider additional Length values not listed in the table. (a) They also get smaller. (b) They remain constant. (c) They get larger, but restricted by some finite maximum. (d) They get larger, unrestricted by any finite maximum. 3. Is it possible to build this rectangle with a Length of zero meters? Briefly explain your answer. 4. Write a formula that computes the Width, y, for a rectangle of Length x. y =. 5. What happens if you try to plug in x = 0 to your formula from #4? Is this consistent with your answer to #3? Definition: The vertical line x = a is a vertical asymptote of the graph of y = f(x) if the values of f (r) approach either co or -co as a approaches a from the left or from the right. 6. Would the graph of your equation from #4 have a vertical asymptote? If so, what is its equation? Explain your answer using the above definition. 20Model 2: Vertical Asymptotes and Holes Open the Desmos activity found at this link. The link is also provided on Canvas. You may choose "Continue without signing in." 7. Write the equation of the blue curve graphed on the first slide. f(x) = 8. Move the slider to set k = -1. The graph should have two vertical asymptotes. Write the equation of each one: I / y - and 9. Write down the formula for f(x) with k = -1 plugged in, then factor it. What relationship is there between the asymptotes and the factors? 10. Find two values of k for which the graph has only one vertical asymptote, then complete these sentences, starting with the lesser of the two k values. . When k = - lesser value -, the vertical asymptote is -, and there is a hole at ~ = cly . When k = -, the vertical asymptote is . and there is a hole at = greater value x/ y 11. For the lesser & value from #10, write down the formula for f(x) with this k value plugged in, then factor and simplify it. What relationship is there among the asymptote, hole, and factors? 12. For the greater k value from #10, write down the formula for f(x) with this k value plugged in, then factor and simplify it. What relationship is there among the asymptote, hole, and factors?Summary: Suppose f(x) = P(x) is a rational function and q(r) = 0. Then r = r is an excluded value for f, and . If x = r causes division by zero in the simplified formula for f(I), then the line x = r is a vertical asymptote of the graph of y = f(x). . If I = r does not cause division by zero in the simplified formula for f(I), then the graph of y = f(x) has a hole at the point where E = r. 13. Check your answers to the previous questions in Model 2 to see if they are consistent with the summary box above. Model 3: Horizontal and Slant Asymptotes 14. Move to the second slide in the Desmos activity. Fill in the table below. Function y1 y2 y3 y4 y5 y6 y7 y8 Degree of numerator Degree of denominator Definition: The horizontal line y = a is a horizontal asymptote of the graph of y = f(r) if the values of f(x) approach a as r approaches either co or -co. 15. The graphs of y1 through ys appear on slides 3-10. Determine whether each graph has a horizontal asymptote. In the table below, write either the equation of the horizontal asymptote, or write "none" if the graph does not have one. Function y1 y2 y3 y4 y5 y6 y7 y8 Hor. asymptote 16. Use #14 and #15 to complete these rules for a rational function f (x) = P(2) + degree n q(x) n, then (b) if m = n, then (c) if m n, then the graph of y = f(r) has horizontal asymptote y = 0. . If m = n, then the graph of y = f(x) has horizontal asymptote y = c, where c is the ratio of the leading coefficients of p and q. . If m <>