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To obtain the graph of [tex]\( f(x) = \frac{1}{x-8} + 14 \)[/tex] from the graph of [tex]\( y = \frac{1}{x} \)[/tex], follow these detailed steps involving transformations:
1. Start with the graph of [tex]\( y = \frac{1}{x} \)[/tex]:
- The graph of [tex]\( y = \frac{1}{x} \)[/tex] is a hyperbola with two branches, one in the first and third quadrants. The asymptotes are the x-axis (horizontal asymptote [tex]\( y = 0 \)[/tex]) and the y-axis (vertical asymptote [tex]\( x = 0 \)[/tex]).
2. Horizontal Shift:
- To apply a horizontal shift, we look at the term inside the function's argument, [tex]\( x - 8 \)[/tex]. This indicates a shift to the right.
- Shift the graph of [tex]\( y = \frac{1}{x} \)[/tex] 8 units to the right:
- This transformation changes each point [tex]\((x, y)\)[/tex] on the graph of [tex]\( y = \frac{1}{x} \)[/tex] to [tex]\((x+8, y)\)[/tex] on the new graph.
- The vertical asymptote, originally at [tex]\( x = 0 \)[/tex], now moves to [tex]\( x = 8 \)[/tex].
- After this shift, the equation becomes [tex]\( y = \frac{1}{x-8} \)[/tex].
3. Vertical Shift:
- To apply a vertical shift, we focus on the term outside the function [tex]\( +14 \)[/tex]. This indicates a shift upwards.
- Shift the graph of [tex]\( y = \frac{1}{x-8} \)[/tex] 14 units upwards:
- This transformation changes each point [tex]\((x, y)\)[/tex] on the graph of [tex]\( y = \frac{1}{x-8} \)[/tex] to [tex]\((x, y+14)\)[/tex] on the new graph.
- The horizontal asymptote, originally at [tex]\( y = 0 \)[/tex], now moves to [tex]\( y = 14 \)[/tex].
- After this shift, the equation becomes [tex]\( y = \frac{1}{x-8} + 14 \)[/tex].
By performing these two transformations, the graph of [tex]\( f(x) = \frac{1}{x-8} + 14 \)[/tex] is obtained from the graph of [tex]\( y = \frac{1}{x} \)[/tex]:
- Step 1: Shift the graph of [tex]\( y = \frac{1}{x} \)[/tex] 8 units to the right to get the graph of [tex]\( y = \frac{1}{x-8} \)[/tex].
- Step 2: Shift the graph of [tex]\( y = \frac{1}{x-8} \)[/tex] 14 units upwards to get the graph of [tex]\( y = \frac{1}{x-8} + 14 \)[/tex].
Thus, the graph of [tex]\( f(x) = \frac{1}{x-8} + 14 \)[/tex] can be obtained by shifting the graph of [tex]\( y = \frac{1}{x} \)[/tex] 8 units to the right and 14 units upward.
1. Start with the graph of [tex]\( y = \frac{1}{x} \)[/tex]:
- The graph of [tex]\( y = \frac{1}{x} \)[/tex] is a hyperbola with two branches, one in the first and third quadrants. The asymptotes are the x-axis (horizontal asymptote [tex]\( y = 0 \)[/tex]) and the y-axis (vertical asymptote [tex]\( x = 0 \)[/tex]).
2. Horizontal Shift:
- To apply a horizontal shift, we look at the term inside the function's argument, [tex]\( x - 8 \)[/tex]. This indicates a shift to the right.
- Shift the graph of [tex]\( y = \frac{1}{x} \)[/tex] 8 units to the right:
- This transformation changes each point [tex]\((x, y)\)[/tex] on the graph of [tex]\( y = \frac{1}{x} \)[/tex] to [tex]\((x+8, y)\)[/tex] on the new graph.
- The vertical asymptote, originally at [tex]\( x = 0 \)[/tex], now moves to [tex]\( x = 8 \)[/tex].
- After this shift, the equation becomes [tex]\( y = \frac{1}{x-8} \)[/tex].
3. Vertical Shift:
- To apply a vertical shift, we focus on the term outside the function [tex]\( +14 \)[/tex]. This indicates a shift upwards.
- Shift the graph of [tex]\( y = \frac{1}{x-8} \)[/tex] 14 units upwards:
- This transformation changes each point [tex]\((x, y)\)[/tex] on the graph of [tex]\( y = \frac{1}{x-8} \)[/tex] to [tex]\((x, y+14)\)[/tex] on the new graph.
- The horizontal asymptote, originally at [tex]\( y = 0 \)[/tex], now moves to [tex]\( y = 14 \)[/tex].
- After this shift, the equation becomes [tex]\( y = \frac{1}{x-8} + 14 \)[/tex].
By performing these two transformations, the graph of [tex]\( f(x) = \frac{1}{x-8} + 14 \)[/tex] is obtained from the graph of [tex]\( y = \frac{1}{x} \)[/tex]:
- Step 1: Shift the graph of [tex]\( y = \frac{1}{x} \)[/tex] 8 units to the right to get the graph of [tex]\( y = \frac{1}{x-8} \)[/tex].
- Step 2: Shift the graph of [tex]\( y = \frac{1}{x-8} \)[/tex] 14 units upwards to get the graph of [tex]\( y = \frac{1}{x-8} + 14 \)[/tex].
Thus, the graph of [tex]\( f(x) = \frac{1}{x-8} + 14 \)[/tex] can be obtained by shifting the graph of [tex]\( y = \frac{1}{x} \)[/tex] 8 units to the right and 14 units upward.
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