Graphs of Increasing Exponential Functions
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1 Section 5 2A: Graphs of Increasing Exponential Functions We want to determine what the graph of an exponential function y = a x looks like for all values of a > We will select a value of a > and examine several ordered pairs for y = a x We will use a = 2 The patterns we find for a = 2 will be true for any value of a > The graph of y = 2 x x and y values for the right side of the graph if x = 0 if x = if x = 2 if x = 3 if x = 4 if x = 5 y = 2 0 y = 2 y = 2 2 y = 2 3 y = 2 4 y = 2 5 y =! y = 2! y = 4! y = 8! y =6! y = 32! As the values for x become larger and larger positive numbers then the values for y become larger and larger positive values. This means that the right end of the graph is a slowly increasing curve pointing up and to the right x and y values for the left side of the graph if x = 6 if x = 5 if x = 4 if x = 3 if x = 2 if x = y = 2 6 y = 2 5 y = 2 4 y = 2 3 y = 2 2 y = 2 y = 64! y = 32! y = 6! y = 8! y = 4! y = 2 As the values for x become larger and larger negative 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 left side of the graph will get closer and closer to the x axis. We call the line that the graph approaches but does not reach an asymptote. We use a dotted line to show the asymptotic line.
2 The table below shows several values for pairs of (x, y) x y The graph of y = 2 x. Domain Range: x 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 x axis. 2. The x intercept is at (, 0 ) 3. Left end of the graph: As the values for x become larger and larger negative 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. We draw the graph of the left end of the graph getting closer and closer to the x axis without touching the x axis. 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. Right end of the graph: As the values for x become larger and larger positive numbers then the values for y become larger and larger positive values. As the values for x increase the y values get extremely large. We show the right end of the graph pointing upwards and to the right and we use an arrow to show that it continues to increase.
3 The Graph of a Increasing Exponential Function The graph of y = a x for all values of a > is shown below. y = a x for all values of a > is called an increasing exponential function. Domain Range: x 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 x axis. 2. The x intercept is at (, 0 ) 3. The x axis is a horizontal asymptote. We use a dotted line to show the asymptotic line. Every increasing logarithmic function has a horizontal asymptote. 3. Left end of the graph: As the values for x become larger and larger negative 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 left side of the graph will get closer and closer to the x axis. We show the graph of the left end of the graph getting closer and closer to the x axis without touching the x axis. 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. Right end of the graph: As the values for x become larger and larger positive numbers then the values for y become larger and larger positive values. As the values for x increase the y values get extremely large. We show the right end of the graph pointing upwards and to the right and we use an arrow to show that it continues to increase.
4 There are 6 different transformation that change the position of the graph, asymptote and y intercept. The graph of y = 2 x and be moved RiGHT, moved LEFT, moved UP, moved Down, flipped about the X 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 = 2 x Translation :! Translation 2:! Translation 3: y = 2 x 3! y = 2 x + 3! y = 2 x + 3! subtracting 3 from x! adding 3 to x! Adding 3 at the end!! moves the graph RIGHT 3! moves the graph LEFT 3! moves the graph UP 3!! The y intercept is found by! The y intercept is found by! The y intercept is found by! letting x = 0 and finding y! letting x = 0 and finding y! letting x = 0 and finding y y = 2 x 3! y = 2 x + 3! y = 2 x + 3 y = 2 0 3! y = ! y = y= 2 3! y = 2 3! y = + 3 y = / 8! y = 8! y = 4!
5 Translating the graph of y = 2 x Translation 4:! Translation 5:! Translation 6: y = 2 x 3! y = 2 x! y = 2 x subtracting 3 at the end! negating y flips the graph! negating x flips the graph moves the graph DOWN 3! about the x axis! about the y axis!! The y intercept is found by! The y intercept is found by! The y intercept is found by! letting x = 0 and finding y! letting x = 0 and finding y! letting x = 0 and finding y y = 2 x 3! y = 2 x! y = 2 x y = 2 0 3! y = 2 0! y = 2 0 y = 3! y =! y = 2 0 y = 2!! y =
6 The graph y = 2 x 3 compared to the graph of y = 2 x y = 2 x 3 is equivalent to y = 2 (x 3) Subtracting 3 from the x inside a bracket moves the graph 3 units to the RIGHT The graph of y = 2 x + 3 compared to the graph of y = 2 x y = 2 x + 3 is equivalent to y = 2 (x + 3) Adding 3 to the x inside a bracket moves the graph 3 units to the LEFT
7 The graph of y = 2 x + 3 compared to the graph of y = 2 x y = 2 x + 3 is equivalent to y = (2 x ) + 3 Adding 3 to the 2 x at the end of the equation moves the graph UP 3 units The graph of y = 2 x + 3 compared to the graph of y = 2 x y = 2 x 3 is equivalent to y = (2 x ) 3 Subtracting 3 to the 2 x at the end of the equation moves the graph DOWN 3 units
8 The graph of y = 2 x compared to the graph of y = 2 x Multiplying 2 x x x by is equivalent to ( ) 2 and is written as y = 2 Multiplying the 2 x by is equivalent to negating the y values (negating the y values flips the graph about the x axis) The graph of y = 2 x compared to the graph of y = 2 x Multiplying the x in the exponent by a negative one has the effect of flipping the graph about the y (negating the x values flips the graph about the y axis)
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