1. Reduce. 2. Intercepts. x x x. L27-Fri-4-Nov-2016-Sec-4-3-Rational-Graphs-Start-Moodle-Q23. Cannot reduce so no holes.
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1 L7-Fri-4-Nov-016-Sec-4-3-Rational-Graphs-Start-Moodle-Q3 8 L7-Fri-4-Nov-016-Sec-4-3-Rational-Graphs-Start-Moodle-Q3 1. Reduce x x x x x Domain: x x 3, x 3 Cannot reduce so no holes.. Intercepts x-intercepts (value of x when y = 0): x 0 x 9 0 x x So, we have the point (-, 0). Touch or cross: Since x + has multiplicity 1, which is odd, graph crosses at (-, 0).
2 L7-Fri-4-Nov-016-Sec-4-3-Rational-Graphs-Start-Moodle-Q3 9 Behavior near x = -: R x x 3x 3 0 variable x g x x So, near (-, 0), function looks like a straight line with negative slope. y-intercept (value of y when x = 0): x x 9 0 y So, we have the point 0, y 9 3. VA These occur when the denominator is 0. So, the vertical asymptotes are the lines x 3 and x 3. The factors (x 3) and (x + 3) have exponents of 1, which is odd, so they have odd multiplicity, so one part of graph goes up and the other down on either side of the VAs. 4. End Behavior HA or OA Since (degree numerator) < (degree denominator) HA is the line y = R(x) = y This is where the graph crosses the HA or OA, if it does. y x 0 x 9 x 0 x So, the graph crosses the horizontal asymptote when x =. The y-value here is R 0 9 We already had this point (-, 0) as the x-intercept.
3 L7-Fri-4-Nov-016-Sec-4-3-Rational-Graphs-Start-Moodle-Q Connect dots and Plot points R(-6) = -0.1 and R(6) = 0.3 As a check, we can see that the function is not even so there is no symmetry about the y axis and it is not odd so there is no symmetry about the origin.
4 L7-Fri-4-Nov-016-Sec-4-3-Rational-Graphs-Start-Moodle-Q Reduce Factor:. Intercepts x-intercepts x x x 5x 6 3 x 1 x 1 Domain. x x 1 Cannot be reduced. x 5x 6 x 1 x 5x 6 0 x 1 0 x 5x 6 0 x x 3 x or x 3 So, we have the points (-, 0) and (-3, 0). Touch or cross: x + has odd multiplicity so graph crosses here. x 3 has odd multiplicity so graph crosses here. Behavior near x = -3: R 3 x 3x x 1 0 variable x g x x So, near (-3, 0), function looks like a straight line with positive slope. Behavior near x = -: x 3x x 1 0 variable x 3 3 R g x 1x 1 1 So, near (-, 0), function looks like a straight line with negative slope. y-intercept y 0 1 So, we have the point (0, 6). 3. VA These occur when the denominator is 0. So, the vertical asymptote is the line x = -1. The factor x + 1 has odd multiplicity so one part of graph goes up and the other down on either side of the line x = -1.
5 L7-Fri-4-Nov-016-Sec-4-3-Rational-Graphs-Start-Moodle-Q End Behavior HA or OA (degree numerator) = (degree denominator) + 1 so use long division to find oblique asymptote: x 4 x x x x x 4x 6 4x 4 The oblique asymptote is y x 4 5. R(x) = y This is where the graph crosses the HA or OA, if it does. y x 5x 6 x 4 x 1 x 5x 6 x 5x This has no solution so the graph does NOT cross the oblique asymptote.
6 L7-Fri-4-Nov-016-Sec-4-3-Rational-Graphs-Start-Moodle-Q Connect dots and Plot points R( 6) =.4 R(3) = 7.5 As a check, we can see that the function is not even so there is no symmetry about the y axis and it is not odd so there is no symmetry about the origin. Look at the beauty of this graph with its curves and ideas of limits and approaching but not touching. Mathematics is a human endeavor and mathematicians can be quite imaginative and romantic. What can you see when you look at the graph? Can you imagine a love story that this graph might represent?
7 L7-Fri-4-Nov-016-Sec-4-3-Rational-Graphs-Start-Moodle-Q Reduce x 1x 1 x x 1 x 1 x 1 x 1 Domain: x x 1, x 1 x 1 x x x 1x 1 1 x 1 x 1 x 1 Q x x 1 Since the (x + 1) factors cancel, there will be a hole at x = 1. To find the y-value of the hole, find Q So, hole is at (-1, 0).. Intercepts. x-intercepts x 1 0 x 1 x 1 So, we have the point (-1, 0). But, as we saw above, this is a hole. Touch or cross: x 1 has multiplicity odd so graph crosses here, at the hole. x 1 x 1 Behavior near x = -1: 0 variable R 1 11 x 1 1 g x x So, near (-1, 0), function looks like a straight line with negative slope. y-intercept x x 1 x 1 y y So, we have the point (0, -1)
8 L7-Fri-4-Nov-016-Sec-4-3-Rational-Graphs-Start-Moodle-Q VA x 1 0 These occur when the denominator is 0. So, the vertical asymptote is the line x=1. x 1 The factor (x 1) has an exponent of 1, which is odd, so it has odd multiplicity, so one part of graph goes up and the other down on either side of the VA. 4. End Behavior HA or OA Degree of p = degree of q so an 1 y 1. b 1 5. R(x) = y This is where the graph crosses the HA or OA, if it does. Original function: y x x 1 1 x 1 x x 1 x 1 x x 1 n But, x = 1 is not in the domain so R does not cross the horizontal asymptote. Alternatively, we can look at the reduced version: Q x y x 1 1 x 1 x 1 x No solution so R does not cross the horizontal asymptote.
9 L7-Fri-4-Nov-016-Sec-4-3-Rational-Graphs-Start-Moodle-Q Connect dots and Plot points R(5) = -3/4 R() = 3
10 L7-Fri-4-Nov-016-Sec-4-3-Rational-Graphs-Start-Moodle-Q3 37 Doppler Effect 1 The Doppler Effect (named after the Austrian physicist Christian Doppler, who proposed it in 184) is the change in frequency of a wave for an observer moving relative to the wave's source. It is heard when a train sounding its horn approaches, passes, and recedes from an observer. The received frequency is higher compared to the emitted frequency during the approach. This is because as the source of the waves is moving toward the observer, each successive wave crest is emitted from a position closer to the observer than the previous wave. Therefore each wave takes slightly less time to reach the observer than the previous wave which means the time between the arrivals of successive wave crests at the observer is reduced, causing an increase in the frequency. We perceive the change in frequency as a change in pitch of the horn (it sounds higher). Similarly, as the source recedes, the observed frequency is lower. s The observed frequency f o of an approaching train is given by f o f s where f s v s is the frequency of the sound at the source, s is the speed of sound in air (770 miles per hour) and v is the speed of the train. 1 Adapted from
11 L7-Fri-4-Nov-016-Sec-4-3-Rational-Graphs-Start-Moodle-Q3 38 For a nice derivation of this formula, see
12 L7-Fri-4-Nov-016-Sec-4-3-Rational-Graphs-Start-Moodle-Q3 39
13 L7-Fri-4-Nov-016-Sec-4-3-Rational-Graphs-Start-Moodle-Q3 40 This makes sense since there is no effect if the source is not moving. The observer and source hear the same frequency of sound. As the velocity of source approaches the velocity of observer the denominator approaches 0 and so the observed frequency approaches infinity. All the wave pulses hit the observer at the same time and all the energy of the sound wave is concentrated so it sounds like a huge explosion of sound. Doppler Effect was used by Edwin Hubble when he determined that distant galaxies were moving away from our own. He looked at the atomic spectra of light coming from the galaxies and noticed that the characteristic patterns were shifted toward the red (longer wavelength and so lower frequency). Using these data he was able to determine the speed at which the galaxies were moving away from us. In 1998 it was discovered that the galaxies were moving away from each other at an accelerating rate! This is due to what is now called dark energy.
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