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To graph the function [tex]\( f(x) = x \)[/tex] after it has been vertically stretched by a factor of 4 and vertically shifted 3 units, follow these steps:
1. Understand the Original Function [tex]\( f(x) = x \)[/tex]:
- This is a simple linear function where the graph is a straight line passing through the origin (0,0) with a slope of 1.
2. Vertical Stretch by a Factor of 4:
- When a function [tex]\( f(x) \)[/tex] is vertically stretched by a factor of 4, the new function is [tex]\( 4f(x) \)[/tex].
- For our function [tex]\( f(x) = x \)[/tex], the new function after vertical stretching is:
[tex]\[ 4f(x) = 4x \][/tex]
- This transformation makes the graph steeper, as each y-value is now 4 times larger for a given x-value.
3. Vertical Shift by 3 Units:
- When a function [tex]\( f(x) \)[/tex] is vertically shifted up by 3 units, the new function is [tex]\( f(x) + 3 \)[/tex].
- For the stretched function [tex]\( 4f(x) = 4x \)[/tex], the new function after the vertical shift is:
[tex]\[ 4x + 3 \][/tex]
- This transformation shifts the entire graph upward by 3 units.
4. Graphing the Transformed Function:
- Plot the transformed function [tex]\( g(x) = 4x + 3 \)[/tex].
- Start with the y-intercept. For [tex]\( g(x) \)[/tex], when [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 4(0) + 3 = 3 \][/tex]
So, the y-intercept is (0, 3).
- Find another point on the line to determine the slope. For example, when [tex]\( x = 1 \)[/tex]:
[tex]\[ y = 4(1) + 3 = 7 \][/tex]
Therefore, the point (1, 7) is also on the line.
- Plot the points (0, 3) and (1, 7) on the graph, and draw a straight line through these points.
5. Check with More Points:
- To ensure accuracy, you can check additional points. For example, when [tex]\( x = -1 \)[/tex]:
[tex]\[ y = 4(-1) + 3 = -1 \][/tex]
Thus, the point (-1, -1) should be plotted as well.
6. Draw the Axes and Label:
- Label the x-axis and y-axis and make sure to mark the points clearly.
- Draw the original function [tex]\( f(x) = x \)[/tex] as a reference (optional for visual comparison).
Here is a visual summary:
- Original Function [tex]\( f(x) = x \)[/tex]:
- Line through (0, 0), (1, 1), (-1, -1), etc.
- Transformed Function [tex]\( g(x) = 4x + 3 \)[/tex]:
- Line through (0, 3), (1, 7), (-1, -1), etc.
By graphing these points and drawing the lines, you will have the graph of the transformed function [tex]\( g(x) = 4x + 3 \)[/tex]. This graph represents the original function [tex]\( f(x) = x \)[/tex] after it has been vertically stretched by a factor of 4 and vertically shifted up by 3 units.
1. Understand the Original Function [tex]\( f(x) = x \)[/tex]:
- This is a simple linear function where the graph is a straight line passing through the origin (0,0) with a slope of 1.
2. Vertical Stretch by a Factor of 4:
- When a function [tex]\( f(x) \)[/tex] is vertically stretched by a factor of 4, the new function is [tex]\( 4f(x) \)[/tex].
- For our function [tex]\( f(x) = x \)[/tex], the new function after vertical stretching is:
[tex]\[ 4f(x) = 4x \][/tex]
- This transformation makes the graph steeper, as each y-value is now 4 times larger for a given x-value.
3. Vertical Shift by 3 Units:
- When a function [tex]\( f(x) \)[/tex] is vertically shifted up by 3 units, the new function is [tex]\( f(x) + 3 \)[/tex].
- For the stretched function [tex]\( 4f(x) = 4x \)[/tex], the new function after the vertical shift is:
[tex]\[ 4x + 3 \][/tex]
- This transformation shifts the entire graph upward by 3 units.
4. Graphing the Transformed Function:
- Plot the transformed function [tex]\( g(x) = 4x + 3 \)[/tex].
- Start with the y-intercept. For [tex]\( g(x) \)[/tex], when [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 4(0) + 3 = 3 \][/tex]
So, the y-intercept is (0, 3).
- Find another point on the line to determine the slope. For example, when [tex]\( x = 1 \)[/tex]:
[tex]\[ y = 4(1) + 3 = 7 \][/tex]
Therefore, the point (1, 7) is also on the line.
- Plot the points (0, 3) and (1, 7) on the graph, and draw a straight line through these points.
5. Check with More Points:
- To ensure accuracy, you can check additional points. For example, when [tex]\( x = -1 \)[/tex]:
[tex]\[ y = 4(-1) + 3 = -1 \][/tex]
Thus, the point (-1, -1) should be plotted as well.
6. Draw the Axes and Label:
- Label the x-axis and y-axis and make sure to mark the points clearly.
- Draw the original function [tex]\( f(x) = x \)[/tex] as a reference (optional for visual comparison).
Here is a visual summary:
- Original Function [tex]\( f(x) = x \)[/tex]:
- Line through (0, 0), (1, 1), (-1, -1), etc.
- Transformed Function [tex]\( g(x) = 4x + 3 \)[/tex]:
- Line through (0, 3), (1, 7), (-1, -1), etc.
By graphing these points and drawing the lines, you will have the graph of the transformed function [tex]\( g(x) = 4x + 3 \)[/tex]. This graph represents the original function [tex]\( f(x) = x \)[/tex] after it has been vertically stretched by a factor of 4 and vertically shifted up by 3 units.
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