Problem 5
Question
Suppose the graph of \(f\) is given. Describe how the graph of each function can be obtained from the graph of \(f\). (a) \(y=f(x)-5\) (b) \(y=f(x-5)\)
Step-by-Step Solution
Verified Answer
(a) The graph shifts 5 units down; (b) The graph shifts 5 units right.
1Step 1: Understanding Vertical Shifts
When the function is in the form of \(y = f(x) - c\), the graph of \(f(x)\) is shifted vertically downwards if \(c\) is positive. This is because each output value of the function is decreased by \(c\), which affects the graph's position on the y-axis.
2Step 2: Applying the Vertical Shift
For the function \(y = f(x) - 5\), we have \(c = 5\). Therefore, the graph of \(f(x)\) is moved 5 units downward. This shift does not affect the shape of the graph; it only changes its position vertically on the coordinate plane.
3Step 3: Understanding Horizontal Shifts
When a function takes the form of \(y = f(x - c)\), the graph of \(f(x)\) is shifted horizontally. Specifically, the graph is moved to the right \(c\) units if \(c\) is positive because each \(x\) value corresponds to a previous \(x - c\) input of \(f\).
4Step 4: Applying the Horizontal Shift
For the function \(y = f(x - 5)\), we have \(c = 5\). Thus, the graph of \(f(x)\) is shifted 5 units to the right. As with the vertical shift, the graph's shape remains unchanged, only its horizontal position is altered.
Key Concepts
Vertical ShiftHorizontal ShiftFunction Graphs
Vertical Shift
A vertical shift involves moving the graph of a function up or down along the y-axis. It is a simple yet fundamental transformation but can greatly affect how a graph appears in relation to the x-axis. Consider a function in the form of \( y = f(x) - c \). Here, \( c \) represents the magnitude of the shift.
When \( c \) is positive, the entire graph moves downward by \( c \) units. Conversely, if \( c \) were negative, the graph would shift upward by \( |c| \) units. This transformation solely changes the vertical placement of the graph. The shape of the graph remains intact.
For example, for \( y = f(x) - 5 \), the graph shifts 5 units downwards. This is because every output value of the function is now 5 less than its original value, resulting in a lower position on the y-axis.
When \( c \) is positive, the entire graph moves downward by \( c \) units. Conversely, if \( c \) were negative, the graph would shift upward by \( |c| \) units. This transformation solely changes the vertical placement of the graph. The shape of the graph remains intact.
For example, for \( y = f(x) - 5 \), the graph shifts 5 units downwards. This is because every output value of the function is now 5 less than its original value, resulting in a lower position on the y-axis.
Horizontal Shift
Horizontal shifts are another type of basic graph transformation. They involve moving the graph left or right along the x-axis. This shift is crucial for adjusting the function’s relation to the time or input axis. The function takes the form \( y = f(x - c) \).
When \( c \) is positive, the graph moves \( c \) units to the right. This happens because the value of \( x \) has to be increased by \( c \) to keep the function's output unchanged. If \( c \) was negative, the graph would shift left by \( |c| \) units. Note that, as with vertical shifts, the shape of the graph does not change; only its horizontal position is altered.
When \( c \) is positive, the graph moves \( c \) units to the right. This happens because the value of \( x \) has to be increased by \( c \) to keep the function's output unchanged. If \( c \) was negative, the graph would shift left by \( |c| \) units. Note that, as with vertical shifts, the shape of the graph does not change; only its horizontal position is altered.
- For example, \( y = f(x - 5) \) indicates a rightward shift of 5 units.
- This adjustment affects where particular points fall along the x-axis, but their relative positions remain the same.
Function Graphs
Function graphs offer a visual representation of mathematical functions, showcasing how one variable affects another. Understanding graph transformations like vertical and horizontal shifts is crucial for accurately interpreting these graphs.
Graphs serve as powerful tools for analysis, allowing you to quickly see characteristics such as intercepts, symmetries, and asymptotes. These traits become evident once you adjust with vertical or horizontal shifts and view the fantastical movements of the graph forms.
Recognizing how a basic graph moves or stretches enables deeper comprehension of more complex functions, settings, and applications in real-world problems.
Graphs serve as powerful tools for analysis, allowing you to quickly see characteristics such as intercepts, symmetries, and asymptotes. These traits become evident once you adjust with vertical or horizontal shifts and view the fantastical movements of the graph forms.
Recognizing how a basic graph moves or stretches enables deeper comprehension of more complex functions, settings, and applications in real-world problems.
- Adjusting for vertical shift changes visibility of features along the y-axis, including maximum and minimum points.
- Meanwhile, horizontal shifts primarily impact the positioning of these features along the x-axis.
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