The parent or reference function for absolute value functions is: \( f(x)=|x| \).
The reference function is the most basic absolute value function, with an \( a \) value of \( 1 \), and the vertex \( (h, k) \) located at the origin \( (0, 0) \).
The reference function is the most basic absolute value function, with an \( a \) value of \( 1 \), and the vertex \( (h, k) \) located at the origin \( (0, 0) \).
When comparing absolute value functions to the reference function, we need to look at what the \( a, h, \) and \( k \) values do to the graph of the functions. Use the applet below to explore the changes that occur between absolute value functions and their graphs. Use the questions below to help guide your exploration.
Use the slider to change the value of \( a \). What happens to the graph of the function as "\( a \)" becomes larger? Smaller? What happens when "\( a \)" is a negative number? Use the slider to change the value of \( h \). How does the graph change as "\( h \)" changes? Use the slider to change the value of \( k \). How does the graph change as "\( k \)" changes?
Comparing Absolute Value Functions
When comparing absolute value functions, there are 3 main ideas to consider. When you are describing the differences between an absolute value function to the reference function, you always want to keep in mind the values of the \( a, h, \) and \( k, \) and the roles they play in the shifts of the graph. Use the following descriptions to compare absolute value functions.
\( a \) value:
Note, the absolute value bars on \( a \) to determine a stretch/compression are not because we are dealing with an absolute value function! This would be needed when analyzing a vertical stretch or compression for any function comparison. You can think of this analysis of simply ''ignoring the sign" on the \( a \) value.
"\( h \)" value:
"\( k \)" value:
Comparing Absolute Value Functions
When comparing absolute value functions, there are 3 main ideas to consider. When you are describing the differences between an absolute value function to the reference function, you always want to keep in mind the values of the \( a, h, \) and \( k, \) and the roles they play in the shifts of the graph. Use the following descriptions to compare absolute value functions.
\( a \) value:
- if \( |a| > 1 \), vertically stretched
- if \( 0 < |a| < 1 \), vertically compressed
- if \( a \) is positive: opens up
- if a is negative: opens down (vertical reflection)
Note, the absolute value bars on \( a \) to determine a stretch/compression are not because we are dealing with an absolute value function! This would be needed when analyzing a vertical stretch or compression for any function comparison. You can think of this analysis of simply ''ignoring the sign" on the \( a \) value.
"\( h \)" value:
- if adding an \( h \) value: vertex translates to the left
- if subtracting an \( h \) value: vertex translates to the right
"\( k \)" value:
- if adding a \( k \) value: vertex translates up
- if subtracting a \( k \) value: vertex translates down
Example 1: Compare the function \( g(x) = 3|x + 5| - 1 \) to the reference function \( f(x) = |x| \).
For \( g(x) = 3|x + 5| - 1 \):
Example 2: Compare the function \( h(x) = \frac{2}{3}|x| + 4 \) to the reference function \( f(x) = |x| \).
For \( h(x) = \frac{2}{3}|x| + 4 \):
Example 3: Compare the function \( k(x) = -2|x - 7| \) to the reference function \( f(x) = |x| \).
For \( k(x) = -2|x - 7| \)
For \( g(x) = 3|x + 5| - 1 \):
- "\( a \)": positive \( 3 \), so the graph would OPEN UP and be STRETCHED since the absolute value of \( 3 \), is greater than \( 1 \)
- "\( h \)": adding a \( 5 \) so the graph would be TRANSLATED to the LEFT \( 5 \) units
- "\( k \)": subtracting a \( 1 \) so the graph would be TRANSLATED DOWN \( 1 \) unit
Example 2: Compare the function \( h(x) = \frac{2}{3}|x| + 4 \) to the reference function \( f(x) = |x| \).
For \( h(x) = \frac{2}{3}|x| + 4 \):
- "\( a \)": positive \( \frac{2}{3} \), so the graph would OPEN UP and be COMPRESSED since the absolute value of \( \frac{2}{3} \), is between \( 0 \) and \( 1 \)
- "\( h \)": there is no \( h \) inside the absolute value so the graph would not be translated to the left or right
- "\( k \)": adding a \( 4 \) so the graph would be TRANSLATED UP \( 4 \) units
Example 3: Compare the function \( k(x) = -2|x - 7| \) to the reference function \( f(x) = |x| \).
For \( k(x) = -2|x - 7| \)
- "\( a \)": negative \( 2 \), so the graph would OPEN DOWN, and be STRETCHED since the absolute value of \( -2 \), is greater than \( 1 \)
- "\( h \)": subtracting an \( h \) so the graph would be TRANSLATED to the RIGHT \( 7 \) units
- "\( k \)": there is no k being added or subtracted after the absolute value so the graph would not be translated up or down
Quick Check
1) Compare the graph of the function \( j(x) = -|x + 4| + 8 \) to the graph of the parent function \( f(x)= |x| \).
2) Write an absolute value function that has the following characteristics: opens up, compressed, and is translated to the left and down.
1) Compare the graph of the function \( j(x) = -|x + 4| + 8 \) to the graph of the parent function \( f(x)= |x| \).
2) Write an absolute value function that has the following characteristics: opens up, compressed, and is translated to the left and down.