QUESTION IMAGE
Question
by the end of this lesson, you will be able to answer the following questions: the graph of function f has a wave - like shape. a graph of y = f(x) follows. (a) complete a table of values. (b) sketch a graph y = |f(x)| and y = f(|x|). (c) describe the similarities and differences among the graphs of y = f(x), y = |f(x)| and y = f(|x|).
Response
Part (a): Complete a table of values
To complete a table of values for \( y = f(x) \), \( y = |f(x)| \), and \( y = f(|x|) \), we first need to identify key points on the graph of \( y = f(x) \). Let's assume the x - axis has integer values (from the graph's grid) and we pick some x - values, say \( x=-3,-2,-1,0,1,2,3 \) (assuming the graph is defined around these points).
- For \( y = f(x) \):
- Observe the y - coordinate of the graph of \( f(x) \) at each x - value. For example, if at \( x = 0 \), the graph of \( f(x) \) crosses the y - axis at \( y=a \) (we can estimate from the wave - like graph), at \( x = 1 \), the y - value is \( b \), etc. Let's create a table:
| \( x \) | \( f(x) \) | \( | f(x) | \) | \( f( | x | ) \) |
|---|---|---|---|---|---|---|---|
| - 2 | \( f(-2) \) | \( | f(-2) | \) | \( f( | -2 | )=f(2) \) |
| - 1 | \( f(-1) \) | \( | f(-1) | \) | \( f( | -1 | )=f(1) \) |
| 0 | \( f(0) \) | \( | f(0) | \) | \( f( | 0 | )=f(0) \) |
| 1 | \( f(1) \) | \( | f(1) | \) | \( f( | 1 | )=f(1) \) |
| 2 | \( f(2) \) | \( | f(2) | \) | \( f( | 2 | )=f(2) \) |
| 3 | \( f(3) \) | \( | f(3) | \) | \( f( | 3 | )=f(3) \) |
- For \( y = |f(x)| \):
- The function \( y = |f(x)| \) takes the absolute value of the output of \( f(x) \). So, if \( f(x)\geq0 \), then \( |f(x)| = f(x) \), and if \( f(x)<0 \), then \( |f(x)|=-f(x) \). For example, if at \( x = - 2 \), \( f(-2)=-k \) (where \( k>0 \)), then \( |f(-2)| = k \).
- For \( y = f(|x|) \):
- The function \( y = f(|x|) \) is an even function. That is, \( f(|-x|)=f(|x|) \), so the graph of \( y = f(|x|) \) is symmetric about the y - axis. For \( x\geq0 \), \( f(|x|)=f(x) \), and for \( x < 0 \), \( f(|x|)=f(-x) \) (since \( |x|=-x \) when \( x < 0 \)). So, the part of the graph for \( x\geq0 \) of \( f(|x|) \) is the same as the graph of \( f(x) \) for \( x\geq0 \), and the part for \( x < 0 \) is the mirror image of the graph of \( f(x) \) for \( x>0 \) about the y - axis.
Part (b): Sketch the graphs
- Graph of \( y = |f(x)| \):
- Start with the graph of \( y = f(x) \). For every point \( (x,y) \) on \( y = f(x) \) where \( y<0 \), reflect that point over the x - axis to get the point \( (x, - y) \) on \( y = |f(x)| \). For points where \( y\geq0 \), the graph of \( y = |f(x)| \) coincides with the graph of \( y = f(x) \).
- Graph of \( y = f(|x|) \):
- Take the graph of \( y = f(x) \) for \( x\geq0 \). Then, reflect this part of the graph over the y - axis to get the graph for \( x < 0 \). The graph of \( y = f(|x|) \) is symmetric about the y - axis. The part of the graph for \( x\geq0 \) is identical to the graph of \( y = f(x) \) for \( x\geq0 \), and the part for \( x < 0 \) is the mirror image of the \( x\geq0 \) part of \( f(x) \) about the y - axis.
Part (c): Similarities and Differences
Similarities:
- Domain (if \( f(x) \) is defined for all real \( x \)): All three functions \( y = f(x) \), \( y = |f(x)| \), and \( y = f(|x|) \) have the same domain (the set of all real numbers for which \( f(x) \) is defined) in terms of the x - values for which the original function \( f(x) \) is defined. For example, if \( f(x) \) is defined for all \( x\in\mathbb{R} \), then all three functions are defined for all \( x\in\mathbb{R} \).
- Intercepts with the y - axis: The y - intercept (when \( x = 0 \)) of all three functions is the same. Because \( f(0)=|f(0)|=f(|0|)=f(0) \), so the point \( (0,f(0)) \) is on all three graphs.
Differences:
- Symmetry:
- The graph of \( y = f(x) \) may o…
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Part (a): Complete a table of values
To complete a table of values for \( y = f(x) \), \( y = |f(x)| \), and \( y = f(|x|) \), we first need to identify key points on the graph of \( y = f(x) \). Let's assume the x - axis has integer values (from the graph's grid) and we pick some x - values, say \( x=-3,-2,-1,0,1,2,3 \) (assuming the graph is defined around these points).
- For \( y = f(x) \):
- Observe the y - coordinate of the graph of \( f(x) \) at each x - value. For example, if at \( x = 0 \), the graph of \( f(x) \) crosses the y - axis at \( y=a \) (we can estimate from the wave - like graph), at \( x = 1 \), the y - value is \( b \), etc. Let's create a table:
| \( x \) | \( f(x) \) | \( | f(x) | \) | \( f( | x | ) \) |
|---|---|---|---|---|---|---|---|
| - 2 | \( f(-2) \) | \( | f(-2) | \) | \( f( | -2 | )=f(2) \) |
| - 1 | \( f(-1) \) | \( | f(-1) | \) | \( f( | -1 | )=f(1) \) |
| 0 | \( f(0) \) | \( | f(0) | \) | \( f( | 0 | )=f(0) \) |
| 1 | \( f(1) \) | \( | f(1) | \) | \( f( | 1 | )=f(1) \) |
| 2 | \( f(2) \) | \( | f(2) | \) | \( f( | 2 | )=f(2) \) |
| 3 | \( f(3) \) | \( | f(3) | \) | \( f( | 3 | )=f(3) \) |
- For \( y = |f(x)| \):
- The function \( y = |f(x)| \) takes the absolute value of the output of \( f(x) \). So, if \( f(x)\geq0 \), then \( |f(x)| = f(x) \), and if \( f(x)<0 \), then \( |f(x)|=-f(x) \). For example, if at \( x = - 2 \), \( f(-2)=-k \) (where \( k>0 \)), then \( |f(-2)| = k \).
- For \( y = f(|x|) \):
- The function \( y = f(|x|) \) is an even function. That is, \( f(|-x|)=f(|x|) \), so the graph of \( y = f(|x|) \) is symmetric about the y - axis. For \( x\geq0 \), \( f(|x|)=f(x) \), and for \( x < 0 \), \( f(|x|)=f(-x) \) (since \( |x|=-x \) when \( x < 0 \)). So, the part of the graph for \( x\geq0 \) of \( f(|x|) \) is the same as the graph of \( f(x) \) for \( x\geq0 \), and the part for \( x < 0 \) is the mirror image of the graph of \( f(x) \) for \( x>0 \) about the y - axis.
Part (b): Sketch the graphs
- Graph of \( y = |f(x)| \):
- Start with the graph of \( y = f(x) \). For every point \( (x,y) \) on \( y = f(x) \) where \( y<0 \), reflect that point over the x - axis to get the point \( (x, - y) \) on \( y = |f(x)| \). For points where \( y\geq0 \), the graph of \( y = |f(x)| \) coincides with the graph of \( y = f(x) \).
- Graph of \( y = f(|x|) \):
- Take the graph of \( y = f(x) \) for \( x\geq0 \). Then, reflect this part of the graph over the y - axis to get the graph for \( x < 0 \). The graph of \( y = f(|x|) \) is symmetric about the y - axis. The part of the graph for \( x\geq0 \) is identical to the graph of \( y = f(x) \) for \( x\geq0 \), and the part for \( x < 0 \) is the mirror image of the \( x\geq0 \) part of \( f(x) \) about the y - axis.
Part (c): Similarities and Differences
Similarities:
- Domain (if \( f(x) \) is defined for all real \( x \)): All three functions \( y = f(x) \), \( y = |f(x)| \), and \( y = f(|x|) \) have the same domain (the set of all real numbers for which \( f(x) \) is defined) in terms of the x - values for which the original function \( f(x) \) is defined. For example, if \( f(x) \) is defined for all \( x\in\mathbb{R} \), then all three functions are defined for all \( x\in\mathbb{R} \).
- Intercepts with the y - axis: The y - intercept (when \( x = 0 \)) of all three functions is the same. Because \( f(0)=|f(0)|=f(|0|)=f(0) \), so the point \( (0,f(0)) \) is on all three graphs.
Differences:
- Symmetry:
- The graph of \( y = f(x) \) may or may not be symmetric (it's a wave - like graph, probably not symmetric about the y - axis or x - axis in general). The graph of \( y = |f(x)| \) is symmetric about the x - axis? No, \( y = |f(x)| \) is always non - negative (since it's an absolute value of a function), and its symmetry is related to the symmetry of \( f(x) \). The graph of \( y = f(|x|) \) is always symmetric about the y - axis (because \( f(|-x|)=f(|x|) \) for all \( x \)).
- Behavior for negative x - values:
- For \( y = f(x) \), the behavior as \( x\) becomes negative is determined by the original wave - like graph. For \( y = |f(x)| \), as \( x\) becomes negative, we are taking the absolute value of the output of \( f(x) \), so any negative parts of \( f(x) \) (for negative \( x \)) are reflected above the x - axis. For \( y = f(|x|) \), as \( x\) becomes negative, \( f(|x|)=f(-x) \), so the graph for \( x < 0 \) is the mirror image of the graph for \( x>0 \) about the y - axis.
- For example, if \( f(x) \) has a "valley" (a local minimum) at \( x=-2 \), for \( y = |f(x)| \), if \( f(-2)<0 \), this valley will be reflected to a "hill" (a local maximum) at \( x = - 2 \). For \( y = f(|x|) \), the point at \( x=-2 \) will have the same y - value as the point at \( x = 2 \) on the graph of \( f(x) \).