We want to determine what the graph of an exponential function. y = a x looks like for all values of a such that 0 > a > 1
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- Lester Holland
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1 Section 5 B: Graphs of Decreasing Eponential Functions We want to determine what the graph of an eponential function y = a looks like for all values of a such that 0 > a > We will select a value of a such that 0 > a > and eamine several ordered pairs We will use a = The patterns we find for a = / will be true for any value of a such that 0 > a > The graph of y = and y values for the right side of the graph if = 0 for y = y = 0 if = for y = y = if = for y = y = if = 3 for y = y = 3 if = for y = y = if = 5 for y = y = 5 y = y = y = y = 8 y = 6 y = 3 As the values for become larger and larger positive values the values for y get closer and closer to 0. These y values will never have a value of 0 but the y values will continue to get closer and closer to 0 as the graph continues to the left. The right side of the graph will get closer and closer to the ais. We call the line that the graph approaches but does not reach an asymptote. We use a dotted line to show the asymptotic line. and y values for the left side of the graph if = 6 if = 5 if = if = 3 if = if = for y = y = y = 6 6 for y = y = y = 3 5 for y = y = y =6 for y = y = y = 8 3 for y = y = y = for y = y = y = As the values for become larger and larger negative numbers then the values for y become larger and larger positive values. This means that the left end of the graph is curve pointing up and to the left
2 The table below shows several values for pairs of (, y) for y = (/) y ( 5,3) y (,6) ( 3,8) (,) (,) (0,) (, ) (3, 8) (5, 3) (7, 8) The graph of y =. Domain Range: can be any real number so the domain of the function is all real numbers. The range for y are positive numbers greater than 0. This means that the entire graph of the function will be above the ais.. The y intercept is at (0, ) 3. Left end of the graph: As the values for become larger and larger negative numbers then the values for y become larger and larger positive values. This means that the left end of the graph is curve pointing up and to the left.. Right end of the graph: As the values for become larger and larger positive values the values for y get closer and closer to 0. These y values will never have a value of 0 but the y values will continue to get closer and closer to 0 as the graph continues to the left. The right side of the graph will get closer and closer to the ais. We call the line that the graph approaches but does not reach an asymptote. We use a dotted line to show the asymptotic line.
3 The Graph of a Decreasing Eponential Function The graph of y = a for all values of a such that 0 > a > is shown below. y = a for all values of a such that 0 > a > is called an decreasing eponential function. Domain Range: can be any real number so the domain of the function is all real numbers. The range for y are positive numbers greater than 0 or y > 0. This means that the entire graph of the function will be above the ais.. The y intercept is at (0, ) 3. The ais is a horizontal asymptote. We use a dotted line to show the asymptotic line. Every increasing logarithmic function has a horizontal asymptote.. Left end of the graph: As the values for become larger and larger negative numbers then the values for y become larger and larger positive values. This means that the left end of the graph is curve pointing up and to the left. 5. Right end of the graph: As the values for become larger and larger positive values the values for y get closer and closer to 0. These y values will never have a value of 0 but the y values will continue to get closer and closer to 0 as the graph continues to the left. The right side of the graph will get closer and closer to the ais. We call the line that the graph approaches but does not reach an asymptote. We use a dotted line to show the asymptotic line.
4 There are 6 different transformation that change the position of the graph, asymptote and y intercept. The graph of y = (/) and be moved RiGHT, moved LEFT, moved UP, moved Down, flipped about the AXIS or flipped about the y AXIS. These transformations are effected by the addition, subtraction or multiplication of various part of the equation. Translating the graph of y = Translation : Translation : Translation 3: y = (/) 3 y = (/) + 3 y = (/) + 3 subtracting 3 from adding 3 to Adding 3 at the end moves the graph RIGHT 3 moves the graph LEFT 3 moves the graph UP 3 8 / 8 3 The y intercept is found by The y intercept is found by The y intercept is found by letting = 0 and finding y letting = 0 and finding y letting = 0 and finding y y = (/) 3 y = (/) + 3 y = (/) + 3 y = (/ ) (0 3) y = (/ ) y = (/) y = (/ ) 3 y = (/ ) + 3 y = + 3 y = 8 y = / 8 y =
5 Translating the graph of y = Translation : Translation 5: Translation 6: y = (/) 3 y = (/) y = (/) subtracting 3 at the end negating y flips the graph negating flips the graph moves the graph DOWN 3 about the ais about the y ais 3 The y intercept is found by The y intercept is found by The y intercept is found by letting = 0 and finding y letting = 0 and finding y letting = 0 and finding y y = (/) 3 y = (/) y = (/) y = (/ ) 0 3 y = (/ ) 0 = y = (/ ) 0 y = 3 y = y = 0 y = y =
6 The graph of y = (/ ) 3 compared to the graph of y = (/ ) y = (/ ) 3 is equivalent to y = (/ ) ( 3) Subtracting 3 from the inside a bracket moves the graph 3 units to the RIGHT compared to y = (/ ) y = (/ ) 3 moves 3 to the right 8 y = (/ ) has a y intercept of y = (/ ) 3 has a y intercept of 8 The graph of y = (/ ) + 3 compared to the graph of y = (/ ) y = + 3 is equivalent to y = ( + 3) Adding 3 to the inside a bracket moves the graph 3 units to the LEFT y = (/ ) + 3 moves 3 to the left compared to y = (/ ) / 8 y = (/ ) has a y intercept of y = (/ ) + 3 has a y intercept of / 8
7 The graph of y = (/ ) + 3 compared to the graph of y = (/ ) y = (/ ) + 3 Adding 3 to the (/ ) at the end of the equation moves the graph UP 3 units y = (/ ) + 3 moves up 3 3 compared to y = (/) 0 y = (/ ) has a y intercept of y = (/ ) + 3 has a y intercept of The graph of y = (/ ) 3 compared to the graph of y = (/ ) Subtracting 3 to the (/ ) at the end of the equation moves the graph DOWN 3 units compared to y = (/ ) y = (/ ) 3 moves down y = (/ ) has a y intercept of y = (/ ) 3 has a y intercept of
8 The graph of y = (/ ) compared to the graph of y = (/ ) Multiplying the (/ ) by is equivalent to negating the y values (negating the y values flips the graph about the ais) compared to y = (/ ) y = (/ ) 0 is flipped about the ais y = (/ ) has a y intercept of y = (/ ) has a y intercept of The graph of y = (/ ) compared to the graph of y = (/ ) Multiplying the in the eponent by a negative one has the effect of flipping the graph about the y (negating the values flips the graph about the y ais) compared to y = (/ ) y = (/ ) is flipped about the y ais y = (/ ) has a y intercept of y = (/ ) has a y intercept of
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