rational functions objective: to graph rational functions without a calculator using what we have...
TRANSCRIPT
Rational Functions
Objective: To graph rational functions without a calculator using
what we have learned.
Things to look for
• When graphing rational functions, there are certain things that we need to know.
1) Intercepts2) Asymptotes3) Increasing/Decreasing4) Concave up/Concave down/inflection points5) Relative extrema
Example 1
• Sketch the graph of 16
822
2
x
xy
Example 1
• Sketch the graph of
1) When x = 0, y = ½2) This function will equal zero when the numerator is
equal to zero. The x-intercepts are (2,0) and (-2,0)
16
822
2
x
xy
)2)(2(2)4(2 2 xxx
Example 1
• Sketch the graph of
3) The vertical asymptotes are the zeros of the denominator which are x = + 4.
4) The horizontal asymptote is the limit of the function as . Here, since the numerator and denominator are the same degree, the horizontal asymptote is y = 2.
16
822
2
x
xy
x
2lim 2
2
2
2 21682
xx
xx
x
Example 1
• Sketch the graph of
• The graph will never cross a vertical asymptote, but may cross a horizontal asymptote. To see if this happens, we set the equation equal to the asymptote and see if there is an x value that will produce the given y value.
16
822
2
x
xy
Example 1
• Sketch the graph of
• The graph will never cross a vertical asymptote, but may cross a horizontal asymptote. To see if this happens, we set the equation equal to the asymptote and see if there is an x value that will produce the given y value.
• This is never true, so it will not cross the asymptote.
16
822
2
x
xy
16
822
2
2
x
x82322 22 xx 832
Example 1
• Sketch the graph of
___+___| ____+____|____-____|___-___ Inc -4 Inc 0 Dec 4 Dec r max sp
16
822
2
x
xy
22
22/
)16(
)2)(82()4)(16()(
x
xxxxxf
2222
33/
)16(
48
)16(
164644)(
x
x
x
xxxxxf
Example 1• Sketch the graph of
______+____|______-______|___+____ c up -4 c down 4 c up
16
822
2
x
xy
22/
)16(
48)(
x
xxf 42
222//
)16(
)16)(4)(48()48()16()(
x
xxxxxf
32
2//
)16(
)4)(48()48)(16()(
x
xxxxf
32
2//
)16(
)316(48)(
x
xxf
Example 1
• Here is what the graph looks like.
___+___| ____+____|____-____|___-___ Inc -4 Inc 0 Dec 4 Dec r max
______+____|______-______|___+____ c up -4 c down 4 c up
Example 3
• Graph 3/2)4( xy
Example 3
• Graph1) y-intercept (0, -42/3)2) X-intercept (4, 0)3) No horizontal or vertical asymptote.
3/2)4( xy
Example 3• Graph1) y-intercept (0, -42/3)2) X-intercept (4, 0)3) No horizontal or vertical asymptote.4) ___-___|___+___ dec 4 inc c.p. (min)5) ___-___|___-___ c d 4 cd cusp
3/2)4( xy
3/1/ )4(3
2)( xxf
3/4// )4(9
2)( xxf
Example 3• Graph1) y-intercept (0, -42/3)2) X-intercept (4, 0)3) No horizontal or vertical asymptote.4) ___-___|___+___ dec 4 inc c.p. (min)5) ___-___|___-___ c d 4 cd cusp
3/2)4( xy
3/1/ )4(3
2)( xxf
3/4// )4(9
2)( xxf
Example 4
• Graph1) y-intercept (0, 0)2) x-intercept (-2, 0) and (0, 0)3) No asymptotes
3/43/1 36 xxy
)2(3360 3/13/43/1 xxxx
Example 4
• Graph1) y-intercept (0, 0)2) x-intercept (-2, 0) and (0, 0)3) No asymptotes
dec inc inc ____-___|___+___|___+___ -1/2 0 min (s.p.) c.p.
3/43/1 36 xxy
)2(3360 3/13/43/1 xxxx
)21(242)( 3/23/13/2/ xxxxxf
Example 4
• Graph1) y-intercept (0, 0)2) x-intercept (-2, 0) and (0, 0)3) No asymptotes
cu cd cu ____+___|___-___|___+___ 0 1 i.p. i.p.
3/43/1 36 xxy
)2(3360 3/13/43/1 xxxx
)1(3
4
3
4
3
4)( 3/53/23/5// xxxxxf
Example 4
• Graph dec inc inc ____-___|___+___|___+___ -1/2 0 min (s.p.) c.p.
cu cd cu ____+___|___-___|___+___ 0 1 i.p. i.p.
3/43/1 36 xxy
Homework
• Section 4.3• Page 264• 1, 3, 15 (ignore the instructions), 25, 33, 35