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To graph the function [tex]\( h(x) = |x - 2| + 5 \)[/tex] using transformations of the basic absolute value function [tex]\( f(x) = |x| \)[/tex], we will follow these steps:
1. Start with the basic absolute value function [tex]\( f(x) = |x| \)[/tex]:
- The graph of [tex]\( f(x) = |x| \)[/tex] is a V-shaped graph that points upwards with its vertex at the origin (0,0).
2. Horizontal Translation:
- The expression inside the absolute value [tex]\( |x - 2| \)[/tex] indicates a horizontal shift.
- The term [tex]\( x - 2 \)[/tex] translates the graph 2 units to the right.
- Mathematically, this means we replace [tex]\( x \)[/tex] with [tex]\( x - 2 \)[/tex] in the function. This results in the function [tex]\( g(x) = |x - 2| \)[/tex].
- The vertex of the graph of [tex]\( g(x) = |x - 2| \)[/tex] is now at (2,0).
3. Vertical Translation:
- After shifting the graph horizontally, we apply the vertical translation by adding 5 to the function [tex]\( g(x) \)[/tex].
- This shifts the entire graph 5 units upwards.
- Thus, the function becomes [tex]\( h(x) = |x - 2| + 5 \)[/tex].
- The vertex of the graph is now at (2,5).
Let's summarize the transformations and their effect on the graph:
- Original function [tex]\( f(x) = |x| \)[/tex] has a vertex at (0,0).
- Shift [tex]\( f(x) = |x| \)[/tex] 2 units to the right to get [tex]\( g(x) = |x - 2| \)[/tex], with a vertex at (2,0).
- Shift [tex]\( g(x) = |x - 2| \)[/tex] 5 units upwards to get [tex]\( h(x) = |x - 2| + 5 \)[/tex], with a vertex at (2,5).
To better understand the graph, we can evaluate [tex]\( h(x) \)[/tex] at several points:
- For [tex]\( x = -3 \)[/tex]:
[tex]\[ h(-3) = |-3 - 2| + 5 = | -5 | + 5 = 5 + 5 = 10 \][/tex]
- For [tex]\( x = -2 \)[/tex]:
[tex]\[ h(-2) = |-2 - 2| + 5 = |-4| + 5 = 4 + 5 = 9 \][/tex]
- For [tex]\( x = -1 \)[/tex]:
[tex]\[ h(-1) = |-1 - 2| + 5 = |-3| + 5 = 3 + 5 = 8 \][/tex]
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ h(0) = |0 - 2| + 5 = | -2 | + 5 = 2 + 5 = 7 \][/tex]
- For [tex]\( x = 1 \)[/tex]:
[tex]\[ h(1) = |1 - 2| + 5 = | -1 | + 5 = 1 + 5 = 6 \][/tex]
- For [tex]\( x = 2 \)[/tex]:
[tex]\[ h(2) = |2 - 2| + 5 = |0| + 5 = 0 + 5 = 5 \][/tex]
- For [tex]\( x = 3 \)[/tex]:
[tex]\[ h(3) = |3 - 2| + 5 = |1| + 5 = 1 + 5 = 6 \][/tex]
- For [tex]\( x = 4 \)[/tex]:
[tex]\[ h(4) = |4 - 2| + 5 = |2| + 5 = 2 + 5 = 7 \][/tex]
- For [tex]\( x = 5 \)[/tex]:
[tex]\[ h(5) = |5 - 2| + 5 = |3| + 5 = 3 + 5 = 8 \][/tex]
Using these points, we can plot the function [tex]\( h(x) = |x - 2| + 5 \)[/tex]:
[tex]\[ (-3, 10), (-2, 9), (-1, 8), (0, 7), (1, 6), (2, 5), (3, 6), (4, 7), (5, 8) \][/tex]
The graph will be a V-shaped graph, opening upwards, with its vertex at (2,5) and symmetrical about the vertical line [tex]\( x = 2 \)[/tex]. This completes the step-by-step graphing of the function [tex]\( h(x) = |x - 2| + 5 \)[/tex].
1. Start with the basic absolute value function [tex]\( f(x) = |x| \)[/tex]:
- The graph of [tex]\( f(x) = |x| \)[/tex] is a V-shaped graph that points upwards with its vertex at the origin (0,0).
2. Horizontal Translation:
- The expression inside the absolute value [tex]\( |x - 2| \)[/tex] indicates a horizontal shift.
- The term [tex]\( x - 2 \)[/tex] translates the graph 2 units to the right.
- Mathematically, this means we replace [tex]\( x \)[/tex] with [tex]\( x - 2 \)[/tex] in the function. This results in the function [tex]\( g(x) = |x - 2| \)[/tex].
- The vertex of the graph of [tex]\( g(x) = |x - 2| \)[/tex] is now at (2,0).
3. Vertical Translation:
- After shifting the graph horizontally, we apply the vertical translation by adding 5 to the function [tex]\( g(x) \)[/tex].
- This shifts the entire graph 5 units upwards.
- Thus, the function becomes [tex]\( h(x) = |x - 2| + 5 \)[/tex].
- The vertex of the graph is now at (2,5).
Let's summarize the transformations and their effect on the graph:
- Original function [tex]\( f(x) = |x| \)[/tex] has a vertex at (0,0).
- Shift [tex]\( f(x) = |x| \)[/tex] 2 units to the right to get [tex]\( g(x) = |x - 2| \)[/tex], with a vertex at (2,0).
- Shift [tex]\( g(x) = |x - 2| \)[/tex] 5 units upwards to get [tex]\( h(x) = |x - 2| + 5 \)[/tex], with a vertex at (2,5).
To better understand the graph, we can evaluate [tex]\( h(x) \)[/tex] at several points:
- For [tex]\( x = -3 \)[/tex]:
[tex]\[ h(-3) = |-3 - 2| + 5 = | -5 | + 5 = 5 + 5 = 10 \][/tex]
- For [tex]\( x = -2 \)[/tex]:
[tex]\[ h(-2) = |-2 - 2| + 5 = |-4| + 5 = 4 + 5 = 9 \][/tex]
- For [tex]\( x = -1 \)[/tex]:
[tex]\[ h(-1) = |-1 - 2| + 5 = |-3| + 5 = 3 + 5 = 8 \][/tex]
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ h(0) = |0 - 2| + 5 = | -2 | + 5 = 2 + 5 = 7 \][/tex]
- For [tex]\( x = 1 \)[/tex]:
[tex]\[ h(1) = |1 - 2| + 5 = | -1 | + 5 = 1 + 5 = 6 \][/tex]
- For [tex]\( x = 2 \)[/tex]:
[tex]\[ h(2) = |2 - 2| + 5 = |0| + 5 = 0 + 5 = 5 \][/tex]
- For [tex]\( x = 3 \)[/tex]:
[tex]\[ h(3) = |3 - 2| + 5 = |1| + 5 = 1 + 5 = 6 \][/tex]
- For [tex]\( x = 4 \)[/tex]:
[tex]\[ h(4) = |4 - 2| + 5 = |2| + 5 = 2 + 5 = 7 \][/tex]
- For [tex]\( x = 5 \)[/tex]:
[tex]\[ h(5) = |5 - 2| + 5 = |3| + 5 = 3 + 5 = 8 \][/tex]
Using these points, we can plot the function [tex]\( h(x) = |x - 2| + 5 \)[/tex]:
[tex]\[ (-3, 10), (-2, 9), (-1, 8), (0, 7), (1, 6), (2, 5), (3, 6), (4, 7), (5, 8) \][/tex]
The graph will be a V-shaped graph, opening upwards, with its vertex at (2,5) and symmetrical about the vertical line [tex]\( x = 2 \)[/tex]. This completes the step-by-step graphing of the function [tex]\( h(x) = |x - 2| + 5 \)[/tex].
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