Section Symmetric across a vertical line called the axis of symmetry that runs through the vertex.

Transcription

1 1 Sectio 3.1 I. Quadratic Fuctio (Parabolas) f ( x) ax bx c, for a 0 Properties of Parabolas: 1. Symmetric across a vertical lie called the axis of symmetry that rus through the vertex.. Vertex- the maximum/miimum poit o the graph. b b Vertex =, f a a 3. Vertical Itercept: (0, f (0)) 4. Cocavity: The graph is cocave up if a > 0 The graph is cocave dow if a < 0 5. Horizotal Itercept- the value of x whe y = 0. There ca be 0, 1, or real horizotal itercepts. Aother ame for horizotal itercept is root or x-itercept. To fid a root: 1. Set f(x) = 0. Solve for x 3. The solutio ca be: No root Oe root: ( r 1,0) Two roots: ( r 1,0) ad ( r,0) Commet: For f ( x) ax bx c, for a 0, if b = c = 0, the a quadratic equatio (parabola) reduces to f ( x) ax.

2 Example: Fid the equatio for a parabola a) What is the y-itercept? b) What are the x-itercepts? c) What is the vertex? d) What is the equatio for the axis of symmetry? e) Fid f(1) ad f(-1). Use these poits to plot g(x) f) What is the domai? g) What is the rage? g( x) ax that rus through the poit (, 8). Commet: I the homework, whe you have to match graphs with the equatios, just look at the vertex ad the cocavity.

3 3 II. Stadard Form (also called Vertex Form) of a Quadratic Fuctio (Parabola) f ( x) a( x h) k, for a 0 Properties for Stadard Form: a 1 stretches the graph 0 a 1 compresses the graph a > 0, graph is cocave up a < 0, graph is cocave dow h > 0, shifts the graph left o the x-axis by h uits. h < 0, shifts the graph right o the x-axis by h uits. k > 0, shifts the graph up o the y-axis by k uits k < 0, shifts the graph dow o the y-axis by k uits Vertex (h, k). To fid a maximum or miimum, fid f(h) = k. Axis of symmetry is at x = h. Y-itercept is at (0, ah k )

4 4 Example A bullet is fired straight ito the air ad has height h( t) a( t 6) 50. a) Estimate the maximum height that the bullet reaches. Whe does this happe? b) Fid the equatio give that the bullet is 30 feet high after secods. c) Where is the axis of symmetry? d) What is the y-itercept? e) Roughly sketch a graph of the equatio.

5 5 III. To covert from f ( x) a( x h) k to f ( x) ax bx c form: 1. Expad the square. Multiply by coefficiet a 3. Add/subtract k 4. Write the terms by descedig powers of the idepedet variable. Example: Write f ( x) 3( x ) 5 i f ( x) ax bx c form. a) Fid the vertex b) What is the maximum? Where does it occur? c) Is the graph cocave up or cocave dow?

6 6 IV. To covert from f ( x) ax bx c to f ( x) a( x h) k form: 1. a is the same for both equatios, so we really just eed to fid h ad k.. Vertex for f ( x) ax bx c is:, b b f a a Vertex for f ( x) a( x h) k is: (h, k) So h b a b ad k = f ( ) a Example: Write fuctio. f ( x) 3x 1x 17 i f ( x) a( x h) k form. Sketch a graph of the

7 7 V. Maximizatio Problems: 1. Amog all pairs of umbers whose sum is 0, fid a pair whose product is as large as possible. What is the maximum product?

8 8. Coldstoe Creamery has a daily cost of $55.00 ad variable costs for makig a bowl of ice cream which are $0.55. a. Let x represet the umber of bowls of ice cream made ad sold each day. Write the daily cost fuctio (C) for Coldstoe. b. The fuctio R( X ).001x 3x describes the moey, i dollars, that Coldstoe takes i each day from the sale of ice cream bowls. Use this reveue fuctio ad the cost fuctio from part a to write the store s weekly profit fuctio, P. c. Use the store s profit fuctio to determie the umber of ice cream bowls that Coldstoe should make ad sell each day to maximize its profit. What is the maximum daily profit?

9 9 Sectio 3. I. Polyomial of degree : f ( x) a x a x... a x a Where is a o-egative iteger represetig the degree of the polyomial a 0, is a coefficiet, which is a costat ax is the leadig term Terms: each separate power fuctio Costat term: a 0 Stadard form: Arrage the terms from the highest power to the lowest power. Example: Special Polyomial Fuctios Liear y = mx + b degree = 1 o bed leadig term: mx Quadratic y ax bx c degree oe bed leadig term: y ax y ax bx y ax c ax Example: Write the polyomial i descedig powers. Idetify the degree ad the leadig term y 5x x x 4x x

10 10 Commet: To get the degree for a polyomial like the oe below, add the expoets of all multiplied terms ivolvig the same base. Ex: Fid the power of the polyomial 3 f ( x) 4 x ( x 1) ( x 5) II. Graphig Polyomials Turig Poit: The umber of times that the graph beds/chages directios A polyomial of degree has at most (-1) turig poits Global Behavior as x approaches ifiity (Leadig Coefficiet Test) The leadig term ax determies the global behavior of the fuctio. ( positive) x odd ( egative) x odd ( positive) x eve ( egative) x eve

11 11 Vertical Itercept the value of y whe x = 0. f ( x) a x a x... a x a f (0) a 0 a 0... a 0 a f(0) a 0 So the vertical itercept is at (0, a 0 ) Horizotal Itercept/Zero/Root the value of x whe y = 0. A polyomial of degree will touch the x-axis at most times To fid the roots, you ca: 1. Estimate them from a graph. Factor the polyomial so that it looks like y a ( x r )( x r ) ( x r ) 1 This polyomial has zeros at x r1, r,..., r Multiplicities If ( x r ) eve the the graph touches the x-axis ad turs aroud at x = r If ( x r ) odd the the graph crosses the x-axis at x = r Overall, the higher the power, the more the graph flattes out.

12 1 Y-axis Symmetry f(-x) = f(x) Origi Symmetry f(-x) = - f(x) Example: Idetify what type of symmetry 4 f ( x) x x 1 Example: Use the graph to aswer the questio below: a) Is the degree of this polyomial eve or odd? b) Idetify the miimum possible degree c) Estimate the vertical ad horizotal itercepts.

13 13 Example: Cosider 3 f ( x) x ( x 1) ( x 5) a) Determie the graph s ed behavior b) Fid the x-itercepts. Discuss. c) Fid the y-itercept. d) What type of symmetry is there? e) Graph the fuctio. f) What are the maximum umber of turig poits?

14 14 Sectio 3.5 Example: Fid each of the followig As x 3, f( x ) As x, f( x ) As x 3, f( x ) As x, f( x )

15 15 I. Recall, a Ratioal Fuctio is writte as f( x) px ( ) qx ( ) Properties of Ratioal Fuctios: 1. Whe q(x) = 0, the graph is ot defied. Case I: We factor the polyomials p(x) ad q(x). Oe of the factors caceled with the umerator. THERE IS A HOLE IN THE GRAPH. Example: f( x) x x 4 Case II: We factor the polyomials p(x) ad q(x). Either othig caceled OR oe of the factors caceled with the deomiator. THERE ARE VERTICAL ASYMPTOTES. Example: Fid the vertical asymptotes, if ay, of the graph of the ratioal fuctio x 3 gx ( ) x 9

16 16. Horizotal Asymptotes : There is at most oe horizotal asymptote for a ratioal equatio. f( x) px ( ) ax a x... a x a d q( x) b x b x... b x b d d 1 1 d If DEGREE OF um < deom, there is a asymptote at y = 0 (the x-axis) If DEGREE OF um = deom, there is a horizotal asymptote at y a b d If DEGREE OF um > deom, there is o horizotal asymptote. But, there is a slat asymptote Fid the slat asymptote usig sythetic divisio. Example: Fid the horizotal asymptote, if ay, for each ratioal fuctio. 15x a) f( x) 3x 1 b) f( x) 15x 3x 1 c) f( x) x x x 3 6

17 17 To fid the slat asymptote, divide q(x) ito p(x): p( x) qx ( ) remaider mx b q( x) The mx+b portio of the aswer is the equatio of the slat asymptote. Example: Fid the slat asymptote of f( x) x x x 3 6

18 18 II. Graphig ratioal fuctios: Method I--Usig trasformatios Start with the basic graphs below, ad the apply the usual trasformatios. Example: Graph the equatio below usig trasformatios. 1 a) hx ( ) 1 ( x 3)

19 19 III. Graphig Ratioal Fuctios: Method II Steps: 1. Determie the symmetry of the graph. Fid the y-itercept 3. Fid the x-itercepts 4. Fid the vertical asymptotes 5. Fid the horizotal asymptote 6. Plot oe poit betwee ad beyod each x-itercept ad vertical asymptote 7. Sketch the graph Example: Graph mx ( ) x x x 1 4

20 0 Example: Graph gx ( ) x x x 3 6 Notice that this graph has a slat asymptote.