More Advanced Parent Functions #2 (Absolute Value, Exponential, and Logarithmic)
Description/Explanation/Highlights
Video Description
This video focuses on the characteristics of the absolute value, exponential growth and decay, and logarithmic parent functions.
Steps and Key Points to Remember
To recognize and describe the characteristics of the absolute value, exponential, and logarithmic parent functions, follow these steps:
- Absolute value, exponential growth and decay, and logarithmic functions are all function families characterized by certain characteristics that start with the simplest form of the function, its parent function.
- Absolute value functions are recognized symbolically by the \(\left|\;\;\right|\) symbols. Its parent function is \(y=\left|x\right|\).
- Since the absolute value parent can use both positive and negative numbers, here are some suggested coordinates we can use to graph the absolute value parent: (-2, 2) (-1, 1) (0, 0) (1, 1) & (2, 2). These are found by plugging the x-values of the coordinate into the parent function to determine the y-values.
- Graphing these points and connecting them with lines gives us the characteristic “V” shape of the absolute value function family.
- The side of the graph left of the y-axis has a slope or -1 and the right side has a slope of 1.
- The vertex (point of the “V”) is at (0, 0).
- The exponential family actually has an infinite number of parents depending on the the value of the base (b). It is represented in general by \(y=b^x\) where b>0 represents an exponential growth function and 0<b<1 represents an exponential decay function.
- We often use \(y=2^x\) to represent the growth parent because 2 is the smallest integer greater than 1 and is easy to graph.
- Coordinates suggested for graphing the exponential growth parent with a base of 2 are: (-3, 1/8) (-2, 1/4) (-1, 1/2) (0, 1) (1, 2) (2, 4) (3, 8).
- The y-values of the coordinates are determined as always by plugging the x-values in the parent. Remember, negative exponents become positive after moving to the denominator of a fraction so: \(2^{-3}=\frac{1}{2^3}=\frac{1}{8}\)
- Note that when the coordinates are graphed and joined by a smooth curve, as x gets smaller (in the direction of negative infinity), the y-values begin to approach 0 but never reach 0. This creates a horizontal asymptote (a line on the graph that the graph approaches but never reaches) at the x-axis.
- As the x-values get larger (in the direction of positive infinity) the Y-values continue to get larger (also in the direction of positive infinity) and no asymptote is created.
- Just as we often use 2 for the base to represent the exponential growth function, we often use 1/2 for the base to represent exponential decay. Again this is done because to represent decay, b must fall between 0 and 1 and 1/2 is the easiest fraction to work with as the base that fits this criteria. So we use \(y=(\frac{1}{2})^x\) as the exponential decay parent function.
- Suggested coordinates (after plugging in x-values to the parent to get Y-values) for graphing the exponential decay parent include: (-3, 8) (-2, 4) (-1, 2) (0, 1) (1, 1/2) (2, 1/4) (3, 1/8).
- Remember, when raising a fraction to a negative power, flip the fraction over and raise it to the same power made positive such as \((\frac{1}{2})^{-3}=2^3=8\).
- Note that just like the exponential growth parent, the graph of the exponential decay parent function also passes through point (0, 1), but unlike the growth parent, the decay parent approaches the asymptote at the x-axis as x gets larger and continues to go to infinity as the x-values get smaller in the direction of negative infinity.
- The parent function of the logarithmic family also has infinite versions depending on the value of the base (b) as represented by \(y=\log_b(x)\).
- We will use a base of 2 to graph the basic shape of the log and the base 2 log parent. So the parent is represented by \(y=\log_2(x)\).
- Remember when creating coordinates for the logarithmic parent that you cannot take the log of negative numbers or zero. Suggested coordinates for graphing include: (1/4, -2) (1/2, -1) (1, 0) (2, 1) (4, 2)
- Just as exponential parents pass through point (0, 1), logarithmic parents pass through point (1, 0) and just as exponential functions approach a horizontal asymptote at the x-axis (y = 0), logarithmic parent functions approach a vertical asymptote at the y-axis from the right side when graphed.
- Since the exponential growth parent and logarithmic parent functions are inverses of each other you will notice that their graphs reflect over the \(y = x\) line.
Here are some key points to keep in mind when identifying characteristics of parent functions and their graph.
- The parent of the absolute value function is: \(y=\left|x\right|\)
- Absolute value functions form the shape of a “V” when graphed.
- The absolute value parent function has a vertex at (0, 0).
- The left side of the graph has a slope of -1 and the right side has a slope of 1.
X
-2
-1
0
1
2
Y
2
1
0
1
2
The table of values to the left is found by substituting the x-values into the \(y=\left|x\right|\) parent function. Graphing these coordinates will give a good representation of the absolute value parent graph and its shape.
- A parent of an exponential growth function for a base of 2 is \(y=2^x\)
- Exponential growth functions have a horizontal asymptote at the x-axis that the graph approaches as the x-values get smaller (in the direction of negative infinity).
- All exponential growth parents pass through the point (0, 1) since any number to the 0 power is equal to 1.
X
-3
-2
-1
0
1
2
3
Y
1/8
1/4
1/2
1
2
4
8
The table of values to the left is found by substituting the x-values into the \(y=2^x\) parent function. Graphing these coordinates will give a good representation of the exponential growth parent graph and its shape.
- A parent of an exponential decay function for a base of 1/2 is \(y=(\frac{1}{2})^x\)
- Exponential decay functions have a horizontal asymptote at the x-axis that the graph approaches as the x-values get larger (in the direction of positive infinity).
- All exponential decay parents (just like growth parents) pass through the point (0, 1) since any number to the 0 power is equal to 1.
X
-3
-2
-1
0
1
2
3
Y
8
4
2
1
1/2
1/4
1/8
The table of values to the left is found by substituting the x-values into the \\(y=(\frac{1}{2})^x\) parent function. Graphing these coordinates will give a good representation of the exponential decay parent graph and its shape.
- The parent of the log base 2 function is \(y=\log_2(x)\)
- Logarithmic parent functions have a vertical asymptote at the y-axis that the graph approaches as the x-values get closer to zero from the right.
- X-values equal to or less than 0 are undefined on the log parent function,
X
1/4
1/2
1
2
4
Y
-2
-1
0
1
2
The table of values to the left is found by substituting the x-values into the \(y=\log_2(x)\) parent function. Graphing these coordinates will give a good representation of the logarithmic parent graph and its shape.
Video Highlights
- 00:00 Introduction
- 00:27 Characteristics of the \(y=\left|x\right|\) absolute value parent function
- 02:27 Characteristics of the \(y=b^x\) exponential parent function
- 03:33 Characteristics of the \(y=2^x\) base 2 exponential growth parent function
- 06:37 Characteristics of the \(y=(\frac{1}{2})^x\) base 1/2 exponential decay parent function
- 09:07 Characteristics of the \(y=\log_2(x)\) base 2 logarithmic parent function
- 13:12 Conclusion
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