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Sketching the graph of an exponential function of the form f ( x ) = b x

Sketch a graph of f ( x ) = 0.25 x . State the domain, range, and asymptote.

Before graphing, identify the behavior and create a table of points for the graph.

  • Since b = 0.25 is between zero and one, we know the function is decreasing. The left tail of the graph will increase without bound, and the right tail will approach the asymptote y = 0.
  • Create a table of points as in [link] .
    x 3 2 1 0 1 2 3
    f ( x ) = 0.25 x 64 16 4 1 0.25 0.0625 0.015625
  • Plot the y -intercept, ( 0 , 1 ) , along with two other points. We can use ( 1 , 4 ) and ( 1 , 0.25 ) .

Draw a smooth curve connecting the points as in [link] .

Graph of the decaying exponential function f(x) = 0.25^x with labeled points at (-1, 4), (0, 1), and (1, 0.25).

The domain is ( , ) ; the range is ( 0 , ) ; the horizontal asymptote is y = 0.

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Sketch the graph of f ( x ) = 4 x . State the domain, range, and asymptote.

The domain is ( , ) ; the range is ( 0 , ) ; the horizontal asymptote is y = 0.

Graph of the increasing exponential function f(x) = 4^x with labeled points at (-1, 0.25), (0, 1), and (1, 4).
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Graphing transformations of exponential functions

Transformations of exponential graphs behave similarly to those of other functions. Just as with other parent functions, we can apply the four types of transformations—shifts, reflections, stretches, and compressions—to the parent function f ( x ) = b x without loss of shape. For instance, just as the quadratic function maintains its parabolic shape when shifted, reflected, stretched, or compressed, the exponential function also maintains its general shape regardless of the transformations applied.

Graphing a vertical shift

The first transformation occurs when we add a constant d to the parent function f ( x ) = b x , giving us a vertical shift     d units in the same direction as the sign. For example, if we begin by graphing a parent function, f ( x ) = 2 x , we can then graph two vertical shifts alongside it, using d = 3 : the upward shift, g ( x ) = 2 x + 3 and the downward shift, h ( x ) = 2 x 3. Both vertical shifts are shown in [link] .

Graph of three functions, g(x) = 2^x+3 in blue with an asymptote at y=3, f(x) = 2^x in orange with an asymptote at y=0, and h(x)=2^x-3 with an asymptote at y=-3. Note that each functions’ transformations are described in the text.

Observe the results of shifting f ( x ) = 2 x vertically:

  • The domain, ( , ) remains unchanged.
  • When the function is shifted up 3 units to g ( x ) = 2 x + 3 :
    • The y- intercept shifts up 3 units to ( 0 , 4 ) .
    • The asymptote shifts up 3 units to y = 3.
    • The range becomes ( 3 , ) .
  • When the function is shifted down 3 units to h ( x ) = 2 x 3 :
    • The y- intercept shifts down 3 units to ( 0 , 2 ) .
    • The asymptote also shifts down 3 units to y = 3.
    • The range becomes ( 3 , ) .

Graphing a horizontal shift

The next transformation occurs when we add a constant c to the input of the parent function f ( x ) = b x , giving us a horizontal shift     c units in the opposite direction of the sign. For example, if we begin by graphing the parent function f ( x ) = 2 x , we can then graph two horizontal shifts alongside it, using c = 3 : the shift left, g ( x ) = 2 x + 3 , and the shift right, h ( x ) = 2 x 3 . Both horizontal shifts are shown in [link] .

Graph of three functions, g(x) = 2^(x+3) in blue, f(x) = 2^x in orange, and h(x)=2^(x-3). Each functions’ asymptotes are at y=0Note that each functions’ transformations are described in the text.

Observe the results of shifting f ( x ) = 2 x horizontally:

  • The domain, ( , ) , remains unchanged.
  • The asymptote, y = 0 , remains unchanged.
  • The y- intercept shifts such that:
    • When the function is shifted left 3 units to g ( x ) = 2 x + 3 , the y -intercept becomes ( 0 , 8 ) . This is because 2 x + 3 = ( 8 ) 2 x , so the initial value of the function is 8.
    • When the function is shifted right 3 units to h ( x ) = 2 x 3 , the y -intercept becomes ( 0 , 1 8 ) . Again, see that 2 x 3 = ( 1 8 ) 2 x , so the initial value of the function is 1 8 .

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
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2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
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can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
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Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
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Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
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A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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