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Factoring Review Factor each completely.

3x 2 -8x+4 11x 2 -99 16x 3 +128. 4. x 3 +2x 2 -4x-8 2x 2 -x-15 10x 3 -80. Factoring Review Factor each completely. (3x-2)(x-2). (x-2)(x+2) 2. (2x+5)(x-3). 11(x+3)(x-3). 16(x+2)(x 2 -2x+4). 10(x-2)(x 2 +2x+4). More Factoring. x 2 – 6x – 16 x 3 – 64 2x 2 – 9x +9

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Factoring Review Factor each completely.

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  1. 3x2-8x+4 11x2-99 16x3+128 4. x3+2x2-4x-8 2x2-x-15 10x3-80 Factoring ReviewFactor each completely. (3x-2)(x-2) (x-2)(x+2)2 (2x+5)(x-3) 11(x+3)(x-3) 16(x+2)(x2-2x+4) 10(x-2)(x2+2x+4)

  2. More Factoring • x2 – 6x – 16 • x3 – 64 • 2x2 – 9x +9 • 5x2 +12x + 4 • 3x2 – 2x – 21 • 3x3 + 81 • x2 – 25 • 2x2 – 8

  3. Domain and Range Domain Range • All of the possible y values • Find the range of • Find the range of • All of the possible x values • Find the domain of • Find the domain of

  4. Zeros X-intercepts Where the graph intersects the x-axis Determine the zeros for the graphs.

  5. End behavior What is happening to the graph as x decreases and as x increases?

  6. Intervals of Increase/Decrease Increasing values (use x) Decreasing values (use x)

  7. Local Extrema • Relative maximum or minimum • Maximum – greatest y-value on some interval of the domain • Minimum – least y-value on some interval of the domain Give the local extrema.

  8. Rational Functions Students will explore rational functions.

  9. Rational Function – a quotient of two polynomial functions Such as:

  10. Steps to graph rational functions • Find the x-intercepts. (Set numerator = 0 and solve the equation) (you may have to factor to solve the equations) • Find vertical asymptote(s). (set denominator = 0 and solve the equation) (you may have to factor to solve the equations)

  11. Steps to Graphing cont. 3. Find horizontal asymptote(s) (HA).3 cases: • If degree of num. < degree of den., then HA is y=0 b. If degree of num = degree of den., then HA is c. If degree of num. > degree of den., then no HA, but there will be a slant asymptote .(will get to later).

  12. Steps to Graphing cont. 4. Graph asymptotes (use dashed lines) 5. Make a T-chart: choose x-values on either side & between all vertical asymptotes. • Graph pts., and connect with curves.

  13. Ex: Graph. State domain & range. • x-intercepts: x=0 • vert. asymp.: x2+1=0 x2= -1 No vert asymp • horiz. asymp: 1<2 (deg. of top < deg. of bottom) y=0 4. x y -2 -.4 -1 -.5 0 0 1 .5 2 .4 (No real solns.)

  14. Domain: all real numbers Range:

  15. Ex: Graph, then state the domain and range. • x-intercepts: 3x2=0 x2=0 x=0 • Vert asymp: x2-4=0 x2=4 x=2 & x=-2 • Horiz asymp: (degrees are =) y=3/1 or y=3 • x y • 4 4 • 3 5.4 • 1 -1 • 0 0 • -1 -1 • -3 5.4 • -4 4 On right of x=2 asymp. Between the 2 asymp. On left of x=-2 asymp.

  16. Domain: all real #’s except -2 & 2 Range: all real #’s except 0<y<3

  17. Ex: Graph, then state the domain & range. • x-intercepts: x2-3x-4=0 (x-4)(x+1)=0 x-4=0 x+1=0 x=4 x=-1 • Vert asymp: x-2=0 x=2 • Horiz asymp: 2>1 (deg. of top > deg. of bottom) no horizontal asymptotes, but there is a slant! • x y • -1 0 • 0 2 • 1 6 • 3 -4 • 4 0 Left of x=2 asymp. Right of x=2 asymp.

  18. Slant Asymptotes • No horizontal asymptotes • To have a slant asymptote, the degree of the numerator is exactly one more than the degree of the denominator. 2 – 1 = 1

  19. Slant asymptotes • Do synthetic division (if possible, divisor x - r); if not, do long division! • The resulting polynomial (ignoring the remainder) is the equation of the slant asymptote. In our example: 2 1 -3 -4 1 -1 -6 Ignore the remainder, use what is left for the equation of the slant asymptote: y=x-1 2 -2

  20. Slant Asymptotes • Long Division

  21. Slant Asymptotes • How do we graph lines (Quick Refresher) • 1. Plot y-intercept on y-axis • 2. From the y-intercept do rise and then run. • 3. Keep repeating rise and run to get more points.

  22. Domain: all real #’s except 2 Range: all real #’s

  23. Point of Discontinuity Occurs when common factors are in the numerator and denominator Example: • We have a common factor (x + 2). So the point of discontinuity is x = -2. (set x + 2 = 0, solve) • The common factor cancels and we’re left with • Now use what we’re left with to find x-intercepts and VA.

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