Graphing an Exponential Function

1. With the rule of the exponential function, replace |x| by a minimum of four values which can be randomly chosen according to the situation.

2. Find the equation of the horizontal asymptote.

3. Place the points obtained |(x,y)| and sketch the asymptote on a Cartesian plane, then connect the points to draw the curve.

Sketch the following exponential function.
||y=2(3)^x||

1. With the rule of the exponential function, replace |x| by a minimum of four values randomly chosen according to the situation

   We can replace |x| by the values |​​0,| |1,| |2,| and |3.|
   ||\begin{align}
   \text{For} \ \ x_1 &= 0, \\
   y_1&=2(3)^{0}\\
   &= 2 \\\\
   \text{For} \ \ x_2 &= 1, \\
   y_2 &= 2(3)^{1}\\
   &= 6\\\\
   \text{For} \ \ x_3 &= 2, \\
   y_3 &= 2(3)^{2} \\
   &= 18 \\\\
   \text{For} \ \ x_4 &= 3, \\
   y_4 &= 2(3)^{3}\\
   &= 54\end{align}||

   Therefore, the following table of values is obtained.

   |​x|	|​0|	|​1|	|​2|	|​3|
   |​y|	|​2|	​|6|	|​18|	|54|​

2. Find the equation of the horizontal asymptote

   In this case, the horizontal asymptote is on the |x|-axis because |k=0.|

3. Plot the points obtained |(x,y)| and the horizontal asymptote on a Cartesian plane. Connect the points to draw the curve

If the exponential function equation is of the form |y=a(c)^{x},| we can immediately find and plot the point |(0,a),| because |a| corresponds to the initial value.

Graph the following exponential function.
||y=4(0.5)^x+2||

1. With the rule of the exponential function, replace |x| with a minimum of four randomly chosen values according to the situation

   For the sake of precision, more than four coordinates of the points of the function can be calculated.

   ||\begin{align}
   \text{For} \ \ x_1 &= -2, \\
   y_1&=4(0.5)^{-2}+2\\
   &= 18 \\\\
   \text{For} \ \ x_2 &= -1, \\
   y_2 &= 4(0.5)^{-1}+2\\
   &= 10\\\\
   \text{For} \ \ x_3 &= 0, \\
   y_3 &= 4(0.5)^{0}+2 \\
   &= 6 \\\\
   \text{For} \ \ x_4 &= 1, \\
   y_4 &= 4(0.5)^{1}+2 \\
   &= 4\\\\
   \text{For} \ \ x_5 &= 2, \\
   y_5 &= 4(0.5)^{2}+2 \\
   &= 3\\\\
   \text{For} \ \ x_6 &= 3, \\
   y_6 &= 4(0.5)^{3}+2 \\
   &= 2{.}5\\\\
   \text{For} \ \ x_7 &= 4, \\
   y_7 &= 4(0.5)^{4}+2 \\
   &= 2.25 \end{align}||

   Next, make a table of values.

   |x​|	|​-2|	|​-1|	|0|​	|1|​	|2|​	|3|​	|4|​
   |y|​	|18|​	|10|​	|6|​	|4|​	|3|​	|2.5|​	|2.25|​

2. Find the equation of the horizontal asymptote

   In this case, the equation of the asymptote is defined by:
   
   ||\begin{align}
   y&= k \\
   y&= 2\end{align}||

If the equation of the exponential function is of the form |y=a(c)^{x}+k,| the equation of the horizontal asymptote |y=k| can be determined. In addition, the point |(0,\ a+k)| can be plotted. In this form, if |x| is replaced with |0,| it leaves |a+k,| which corresponds to the initial value of the function.

Graph the following exponential function.
||y = -1(2)^{4(x-2)}+5||

1. With the rule of the exponential function, replace |x| with a minimum of four values ​​randomly chosen according to the situation

   We can replace |x| with the values |2,\ 3,\ 4| and |5.|

   ||\begin{align}
   \text{For} \ \ x_1 &= 2, \\
   y_1&=-1(2)^{4(2-2)}+5\\
   &= 4 \\\\
   \text{For} \ \ x_2 &= 3, \\
   y_2 &= -1(2)^{4(3 - 2)} + 5\\
   &= -11\\\\
   \text{For} \ \ x_3 &= 4,\\
   y_3 &= -1(2)^{4(4 - 2)} + 5\\
   &= -251 \\\\
   \text{For} \ \ x_4 &= 5,\\
   y_4 &= -1(2)^{4(5 - 2)} + 5\\
   &= -4\ 091\end{align}||

   The following table of values is obtained.

   ​|x|	|2|​	​|3|	|4|​	|5|​
   |y|​	|4|​	|-11|​	|-251|​	|-4\ 091|​

2. Find the equation of the horizontal asymptote

   Here, the equation of the asymptote is defined by:

   
   ||\begin{align}
   y&= k \\
   y&= 5\end{align}||

1. Sketch the basic exponential function |y=(c)^x.|

2. If necessary, modify the vertical scale change created by parameter |a| and the reflection.

3. If necessary, modify the horizontal scale change created by parameter |b| (factor |\dfrac{1}{{\mid}b{\mid}}|) and the reflection.

4. Perform the vertical translation indicated by the parameter |k.|

5. Perform the horizontal translation indicated by the parameter |h.|

Note: These last four operations can be performed in any order.

Graph the following exponential function.
||y=2(2)^{-3(x+4)}-3||

1. Sketch the basic exponential function |y=(c)^x|
   
   In this example,

   ||\begin{align}
   y&=(c)^x \\
   y&= (2)^x \end{align}||

2. If necessary, modify the vertical scale change created by parameter |a| and the reflection

   Since parameter |a| is equal to |2,| the curve must be "stretched" vertically by a factor of |2.| This indicates that it is necessary to multiply all of the |y|-values of the basic function by |2.|

3. If necessary, modify the horizontal scale change created by the parameter |b| (factor |\frac{1}{\mid b \mid}|) and reflection

   Since parameter |b| is equal to |-3,| the curve must be reflected with respect to the |y|-axis. The curve must also be “contracted” horizontally by a factor of |\frac{1}{3}.| All of the |x|-values of the basic function must be divided by |-3.|

4. Perform the vertical translation indicated by the parameter |k|

   Since parameter |k| is equal to |-3,| a vertical translation of |3| units downwards is required.

5. Perform the horizontal translation indicated by the parameter |h|

   Since parameter |h| is equal to |-4,| |\big(x-(-4)\big),| a horizontal translation of |4| units to the left is required.

Therefore, the desired curve is obtained.

The characteristics of the curve obtained can be verified:

• a horizontal asymptote at |y = -3| is obtained which corresponds to |y = k;|

• a decreasing curve is obtained whereby the values of |y| are greater than |k,| as is the case when parameter |a| is positive and parameter |b| is negative.

To ensure that the graph sketched is correct, here are some important things to verify:

• The position of the asymptote (given by the parameter |k|);

• The reflections in relation to the two axes (given by the signs of the parameters |a| and |b|);

• The direction of the function (if it’s increasing or decreasing).