1. BMayer@ChabotCollege.edu MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §J Graph Rational Fcns

2. BMayer@ChabotCollege.edu MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §5.7 → PolyNomical Eqn Applications  Any QUESTIONS About HomeWork §5.7 → HW-21 5.7 MTH 55

3. BMayer@ChabotCollege.edu MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt 3 Bruce Mayer, PE Chabot College Mathematics GRAPH BY PLOTTING POINTS  Step1. Make a representative T-table of solutions of the equation.  Step 2. Plot the solutions as ordered pairs in the Cartesian coordinate plane.  Step 3. Connect the solutions in Step 2 by a smooth curve

4. BMayer@ChabotCollege.edu MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt 4 Bruce Mayer, PE Chabot College Mathematics Making Complete Plots 1.Arrows in POSITIVE Direction Only 2.Label x & y axes on POSITIVE ends 3.Mark and label at least one unit on each axis 4.Use a ruler for Axes & Straight-Lines 5.Label significant points or quantities           

5. BMayer@ChabotCollege.edu MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt 5 Bruce Mayer, PE Chabot College Mathematics Rational Function  A rational function is a function f that is a quotient of two polynomials, that is,  Where where p(x) and q(x) are polynomials and where q(x) is not the zero polynomial. The domain of f consists of all inputs x for which q(x) ≠ 0.

6. BMayer@ChabotCollege.edu MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt 6 Bruce Mayer, PE Chabot College Mathematics Visualizing Domain and Range  Domain = the set of a function’s Inputs, as found on the horizontal axis (the x-Axis)  Range = the set of a function’s OUTputs, found on the vertical axis (the y-Axis).

7. BMayer@ChabotCollege.edu MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt 7 Bruce Mayer, PE Chabot College Mathematics Find Rational Function Domain 1.Write an equation that sets the DENOMINATOR of the rational function equal to 0. 2.Solve the equation. 3.Exclude the value(s) found in step 2 from the function’s domain.

8. BMayer@ChabotCollege.edu MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt 8 Bruce Mayer, PE Chabot College Mathematics Example  Domain & Range  Graph y = f(x) = x 2. Then State the Domain & Range of the function  Select integers for x, starting with −2 and ending with +2. The T-table:

9. BMayer@ChabotCollege.edu MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt 9 Bruce Mayer, PE Chabot College Mathematics Example  Domain & Range  Now Plot the Five Points and connect them with a smooth Curve (−2,4)(2,4) (−1,1)(1,1) (0,0)

10. BMayer@ChabotCollege.edu MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt 10 Bruce Mayer, PE Chabot College Mathematics Example  Domain & Range  The DOMAIN of a function is the set of all first (or “x”) components of the Ordered Pairs.  Projecting on the X-axis the x-components of ALL POSSIBLE ordered pairs displays the DOMAIN of the function just plotted

11. BMayer@ChabotCollege.edu MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt 11 Bruce Mayer, PE Chabot College Mathematics Example  Domain & Range  Domain of y = f(x) = x 2 Graphically  This Projection Pattern Reveals a Domain of

12. BMayer@ChabotCollege.edu MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt 12 Bruce Mayer, PE Chabot College Mathematics Example  Domain & Range  The RANGE of a function is the set of all second (or “y”) components of the ordered pairs. The projection of the graph onto the y-axis shows the range

13. BMayer@ChabotCollege.edu MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt 13 Bruce Mayer, PE Chabot College Mathematics Domain Restrictions  EVERY element, x, in a functional Domain MUST produce a VALID Range output, y  ReCall the Real-Number Operations that Produce INvalid Results Division by Zero Square-Root of a Negative Number  x-values that Produce EITHER of the above can NOT be in the Function Domain

14. BMayer@ChabotCollege.edu MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt 14 Bruce Mayer, PE Chabot College Mathematics Example  find Domain:  SOLUTION  Avoid Division by Zero Set the DENOMINATOR equal to 0. Factor out the monomial GCF, y. Use the zero-products theorem.  The function is UNdefined if y is replaced by 0, −4, or −1, so the domain is {y|y ≠ −4, −1, 0} FOIL Factor by Guessing Solve the MiniEquations for y

15. BMayer@ChabotCollege.edu MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt 15 Bruce Mayer, PE Chabot College Mathematics Example   Find the DOMAIN and GRAPH for f(x)  SOLUTION When the denom x = 0, we have a Div-by-Zero, so the only input that results in a denominator of 0 is 0. Thus the domain {x|x  0} or (– , 0) U (0,  )  Construct T-table  Next Plot points & connect Dots

16. BMayer@ChabotCollege.edu MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt 16 Bruce Mayer, PE Chabot College Mathematics Plot  Note that the Plot approaches, but never touches, the y-axis (as x ≠ 0) –In other words the graph approaches the LINE x = 0 the x-axis (as 1/   0) –In other words the graph approaches the LINE y = 0  A line that is approached by a graph is called an ASYMPTOTE

17. BMayer@ChabotCollege.edu MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt 17 Bruce Mayer, PE Chabot College Mathematics Vertical Asymptotes  The VERTICAL asymptotes of a rational function f(x) = p(x)/q(x) are found by determining the ZEROS of q(x) that are NOT also ZEROS of p(x). If p(x) and q(x) are polynomials with no common factors other than constants, we need to determine only the zeros of the denominator q(x).  If a is a zero of the denominator, then the Line x = a is a vertical asymptote for the graph of the function.

18. BMayer@ChabotCollege.edu MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt 18 Bruce Mayer, PE Chabot College Mathematics Example  Vertical Asymptote  Determine the vertical asymptotes of the function  Factor to find the zeros of the denominator: x 2 − 4 = 0 = (x + 2)(x − 2)  Thus the vertical asymptotes are the lines x = −2 & x = 2

19. BMayer@ChabotCollege.edu MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt 19 Bruce Mayer, PE Chabot College Mathematics Horizontal Asymptotes  When the numerator and the denominator of a rational function have the same degree, the line y = a/b is the horizontal asymptote, where a and b are the leading coefficients of the numerator and the denominator, respectively.  In This case The line y = c = a/b is a horizontal asymptote.

20. BMayer@ChabotCollege.edu MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt 20 Bruce Mayer, PE Chabot College Mathematics Example  Horiz. Asymptote  Find the horizontal asymptote for  The numerator and denominator have the same degree. The ratio of the leading coefficients is 6/9, so the line y = 2/3 is the horizontal asymptote

21. BMayer@ChabotCollege.edu MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt 21 Bruce Mayer, PE Chabot College Mathematics Finding a Horizontal Asymptote  When the numerator and the denominator of a rational function have the same degree, the line y = a/b is the horizontal asymptote, where a and b are the leading coefficients of the numerator and the denominator, respectively.  When the degree of the numerator of a rational function is less than the degree of the denominator, the x-axis, or y = 0, is the horizontal asymptote.  When the degree of the numerator of a rational function is greater than the degree of the denominator, there is no horizontal asymptote.

22. BMayer@ChabotCollege.edu MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt 22 Bruce Mayer, PE Chabot College Mathematics Asymptotic Behavior  The graph of a rational function never crosses a vertical asymptote  The graph of a rational function might cross a horizontal asymptote but does not necessarily do so

23. BMayer@ChabotCollege.edu MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt 23 Bruce Mayer, PE Chabot College Mathematics Example  Graph  SOLUTION  Vertical asymptotes: x + 3 = 0, so x = −3  The degree of the numerator and denominator is the same. Thus y = 2 is the horizontal asymptote  Graph Plan Draw the asymptotes with dashed lines. Compute and plot some ordered pairs and connect the dots to draw the curve.

24. BMayer@ChabotCollege.edu MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt 24 Bruce Mayer, PE Chabot College Mathematics Example  Graph  Construct T-Table 4/52 00 44 22 8 44 5 55 3.5 77 h(x)h(x)x  Plot Points, “Dash In” Asymptotes

25. BMayer@ChabotCollege.edu MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt 25 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §J1 Exercise Set J2, J4, J6  Watch the DENOMINATOR PolyNomial; it can Produce Div-by-Zero

26. BMayer@ChabotCollege.edu MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt 26 Bruce Mayer, PE Chabot College Mathematics All Done for Today Asymptote Architecture wins competition for WBCB Tower, to be tallest building in Asia

27. BMayer@ChabotCollege.edu MTH55_Lec-06_sec_1-3_Graph_Functions.ppt.ppt 27 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –