`Three sides of a 500 square foot rectangle are fenced. Express the fence s length f as a function of height x.

Transcription

1 Math 140 Lecture 9 See inside text s front cover for area and volume formulas Classwork, remember units Don t just memorize steps, try to understand instead If you understand, every test problem will be like a prevous problem Classwork 10 is hard, work on it before class Exam 1 this week `Three sides of a 500 square foot rectangle are fenced Express the fence s length f as a function of height x x y x Area = 500 = length of fence f `Three sides of a 500 square foot rectangle are fenced Express the fence s length f as a function of width x y f, in feet, = x y Area = 500 = length of fence x 500 x 2x 500 x x 1000 x 2x 1000 x (A) (B) (C) (D) (E) # f

2 Graphing polynomials A polynomial graph is smooth: no breaks, no sharp corners smooth no sharp corners no breaks

3 lead term degree constant 8x 4 3x 2 20 lead coefficient `y x 2 x 4 `y x 2 5 3x 4 Leading term =? Throw away terms not of highest degree (A) -1 (B) 1 (C) (D) (E) x 2 x 4 x 4 Degree =? (A) -1 (B) 0 (C) 1 (D) 2 (E) 4 Constant term =? Equals y-intercept, to get it, set x equal to 0 (A) -1 (B) 0 (C) 1 (D) 2 (E) 4 Continuing problem: Graph the following ` ` y x 2 3 2x x y x 2 3 2x 1 2 x 2 Draw large x and y axes Mark each result on the graph

4 w To get the constant term (y-intercept) for factored polynomials, set x = 0 Find the y-intercept Mark the intercept on your graph ` y x 2 3 2x x ` y x 2 3 2x 1 2 x 2 (A) -32 (B) -16 (C) -8 (D) 8 (E) 16 16

5 y= x y= x 3 y= x x 3, x 5, x 7, x 9, x 2, x 4, x 6, x 8, x 5, x 7, x 9, look like x 3, x 4, x 6, x 8, look like x 2 wfor large x, as you move far from the origin (toward +5 or -5), the graph looks like the leading term ax n For positive a, ax n looks like x n top row -ax n looks like -x n bottom row

6 w To get the leading term of a factored polynomial, replace each factor by its leading term ` y x 2 3 2x 1 2 x 2 ` y x 2 3 2x x `Find the leading term 4x 6 (A) 4x 6 (B) 2x 6 (C) 2x 6 (D) 4x 6 (E) # Put direction arrows on your graph `Find the shape of the graph for large x (A) y= x y= x 3 (B) y= x 2 (C) (D) 3 (E) 2

7 At a root, the graph looks like the graph of root s factor At degrees >2, the tangent at the root is horizontal wat roots of degree 1, the sign (+ or -) changes, the graph crosses x-axis like y = x or y = -x y= x or wat roots of odd degrees 3, 5, 7,, the sign changes, the graph crosses the x-axis like y = x 3 or y = -x 3 y= x 3 3 or wat roots of even degrees 2, 4, 6,, no sign change, the graph touches but does not cross the x-axis, like y = x 2 or y = -x 2 The sign is the same on both sides y= x 2 or 2

8 Use the factored form to get the key numbers and their degrees ` y x 2 3 2x 1 2 x 2 ` y x 2 3 2x x y-intercept: y-intercept: Lead term: Lead term: x 3 2x 2 x 4x 6 1 Key numbers: x 2, 2,2 For me, the degrees are 3, 2, 1 Key intervals:, 2, 2, 1 2, 1 2,2, 2, Sign Rule: At key numbers (where 0 or undefined): If the degree is odd, the sign changes If the degree is even, the sign does not change The y-intercept gives the sign of the interval containing 0 The leading term gives the sign of the first and last intervals The sign rule determines the remaining key interval signs `Find the pattern of signs on the 4 key intervals, 2, 2, 1 2, 1 2,2, 2, (A) (B) (C) (D) (E) # Cross off (mentally or with light pencil strokes) the regions where the graph can not be

9 ` y x 2 3 2x 1 2 x 2 For the first factor, find the root, the degree and the shape x 2 3 Factor: : `Root x = `Degree of root = `Find the shape of the graph around this root y= x (A) or (B) (C) y= x 3 3 (D) y= x 2 (E) 2 Mark the correct shape on your graph

10 ` y x 2 3 2x 1 2 x 2 For the second factor 2x 1, the root is x = ½ and the degree is 1 `Find the shape of the graph around this root Mark it (A) y= x (B) (C) (D) y= x 3 3 y= x 2 (E) or 2 Put the shape on the graph Where the graph is positive / negative determines whether the root looks like or like x n x n

11 ` y x 2 3 2x 1 2 x 2 For the third factor 2 x 2, the root is 2 and the degree is 2 `Find the shape of the graph around this root y= x (A) or (B) (C) y= x 3 3 (D) y= x 2 (E) 2

12 `` y x 2 3 2x 1 2 x 2 Connect the shapes and draw the graph

13 When graphing: w Find the y-intercept Mark it on the graph w Find the leading term Mark its directions on the graph w Find the roots (x-intercepts) and their degrees Mark on graph For classwork and homework: w Calculate and plot a key value in each key interval On exams: w Determine where the graph is positive/negative Cross off (either mentally or very lightly) negative areas below positive key intervals; cross off positive areas above negative intervals b

14 Lecture 10 s iclicker quiz and classwork are hard Study in advance DEFINITION A rational function is a ratio of two polynomials x 2 4 Eg, x 2 4x 4 It is reduced if the top and bottom have no common factors x 2 4 x 2 4x 4 x 2 x 2 x 2 x 2 x 2 x 2 reduced Like polynomials, rational functions have smooth graphs But they may have asymptotes horizontal asymptote vertical asymptote vertical asymptote Graphs can cross horizontal asymptotes, but not vertical bb