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Worked solution

Reflect a rational graph in the y-axis

State the transformation which transforms y=1+1/(x+1)y=1+1/(x+1) to y=1+1/(1−x)y=1+1/(1-x). On separate diagrams, sketch the two graphs, stating the equations of any asymptotes.

CJC · Transformations · Example 6

State the reflection

Answer

Reflection in the y-axis.

Replace x by −x-x: y=1+1/((−x)+1)=1+1/(1−x)y=1+1/((-x)+1)=1+1/(1-x). The Teacher supplies this intermediate substitution and transformation.

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Sketch separately and transform the asymptotes

Answer

Original: x=−1x=-1, y=1y=1. Reflected: x=1x=1, y=1y=1.

Keep both branches on each separate diagram. The source derives the new vertical asymptote from x=−1x=-1 by replacement: −x=−1-x=-1, hence x=1x=1. The horizontal asymptote y=1y=1 is independent of x and remains unchanged.

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Separate two-branch graphs of 1+1/(x+1) and 1+1/(1−x). Vertical asymptote moves from x=−1 to x=1; horizontal asymptote stays y=1 under reflection in the y-axis.
Separate two-branch graphs of 1+1/(x+1) and 1+1/(1−x). Vertical asymptote moves from x=−1 to x=1; horizontal asymptote stays y=1 under reflection in the y-axis.
Separate two-branch graphs of 1+1/(x+1) and 1+1/(1−x). Vertical asymptote moves from x=−1 to x=1; horizontal asymptote stays y=1 under reflection in the y-axis.
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