Factoring Practice. 1. x x 2 10x x x x x 3-8. (x 4)(x + 4) (x 6)(x 4) (x + 3)(x 2-3x + 9)

Transcription

1 Factoring Practice 1. x 2 16 (x 4)(x + 4) 4. x 2 10x + 24 (x 6)(x 4) 2. x (x + 3)(x 2-3x + 9) 5. 16x (2x 3)(2x + 3) 3. 25x x 3-8 5(5x 2 + 3) (3x 2)(9x 2 +6x + 4)

2 5.2 Graphing Simple Rational Functions p. 310 What is the general form of a rational function? What does the h & k tell you? What does the graph of a hyperbola look like? What does the graph of (ax+b)/(cx+d) tell you? What information does the domain & range tell you?

3 Rational Function A function of the form where p(x) & q(x) are polynomials and q(x) 0.

4 Hyperbola A type of rational function. Has 1 vertical asymptote and 1 horizontal asymptote. Has 2 parts called branches. (blue parts) They are symmetrical. We ll discuss 2 different forms. x=0 y = 1/x y=0

5 Hyperbola Form 1 First form: Has 2 asymptotes: x=h (vert.) and y=k (horiz.) Graph 2 points on the right side of the vertical asymptote and 2 points on the left side of the vertical asymptote. Draw the branches.

6 Graph the equation -4 y = -1 x+2 Find asymptotes, graph 4 points, draw branches Vertical: x = h x = -2 Horizontal: y = k y = -1 Try x = {-4,-3,-1,0}

7 Graph the equation y = 10 x Find asymptotes, graph 4 points, draw branches

8 Second form: Hyperbola Form 2 Vertical asymptote: Look at the denominator. Set it equal to 0 and solve for x. Horizontal asymptote: Graph 2 points to the left and 2 points to the right of the horizontal asymptote. Draw the 2 branches.

9 Graph the equation y = 2x+1 x-3 Find asymptotes, graph 4 points, draw branches Vertical: x -3 = 0 x = 3 Horizontal: y = a/c y = 2 Try x = {0,2,4,6}

10 Graph the equation y = x-1 x + 5 Find asymptotes, graph 4 points, draw branches

11 6 Graph the function y = x. Compare the graph with the graph of y = 1 x. SOLUTION STEP 1 Draw the asymptotes x = 0 and y = 0. STEP 2 Plot points to the left and to the right of the vertical asymptote, such as ( 3, 2), ( 2, 3), (2, 3), and (3, 2).

12 STEP 3 Draw the branches of the hyperbola so that they pass through the plotted points and approach the asymptotes. 6 The graph of y = x lies farther from the axes than the graph of y = 1. x Both graphs lie in the first and third quadrants and have the same asymptotes, domain, and range.

13 Domain and Range? Domain: All real numbers except the vertical asymptote. Range: All real numbers except the horizontal asymptote. That makes sense right? Our graphs get super super close to the asymptote but never touch them and both halves of the graph approach both asymptotes.

14 Ex: Graph State the domain & range. Vertical Asymptote: x=1 Horizontal Asymptote: y=2 x y Left of vert. asymp. Right of vert. asymp. Domain: all real # s except 1. Range: all real # s except 2.

15 Ex: Graph State domain & range. Vertical asymptote: 3x+3=0 (set denominator =0) 3x=-3 x= -1 x y Horizontal Asymptote: Domain: All real # s except -1. Range: All real # s except 1/3.

16 W/O graphing, state the domain and range. 1 y = + 2 x-3 D: x 3 R: y 2 y = 1 x+2 D: x -2 R: y 0 y = 6x-1 3x-1 y = x+6 4x-8 D: x 1/3 R: y 2 D: x 2 R: y 1/4

17 3-D Modeling A 3-D printer builds up layers of material to make three dimensional models. Each deposited layer bonds to the layer below it. A company decides to make small display models of engine components using a 3-D printer. The printer costs $24,000. The material for each model costs $300. Write an equation that gives the average cost per model as a function of the number of models printed.

18 Graph the function. Use the graph to estimate how many models must be printed for the average cost per model to fall to $700. What happens to the average cost as more models are printed? SOLUTION STEP 1 Write a function. Let c be the average cost and m be the number of models printed. Unit cost Number printed + Cost of printer c = Number printed = 300m + 24,000 m

19 STEP 2 Graph the function. The asymptotes are the lines m = 0 and c = 300. The average cost falls to $700 per model after 60 models are printed. STEP 3 Interpret the graph. As more models are printed, the average cost per model approaches $300.

20 Graph the function. State the domain and range. y = x 1 x + 3 ANSWER domain: all real numbers except 3, range: all real numbers except 1.

21 What is the general form of a rational function? What does the h & k tell you? Asymptotes are x = h, y = k What does the graph of a hyperbola look like? Two symmetrical branches in opposite quadrants. What does the graph of (ax+b)/(cx+d) tell you? cx+d = 0 is the vertical asymptote and y = a/c is the horizontal asymptote What information does the domain & range tell you? Domain tells what numbers can be used for x and the range is the y numbers when put into the equation.

22 Assignment p. 313, 6-8, 14-20, 28-31