Porisma

Worked solution

Reflect a rational graph in the x-axis

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

CJC · Transformations · Example 5

State the reflection

Answer

Reflection in the x-axis.

Replace y by −y-y: −y=1+1/(x+1)-y=1+1/(x+1), giving y=−1−1/(x+1)y=-1-1/(x+1). The Teacher supplies this intermediate equation 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 in each separate diagram. The source derives the new horizontal asymptote by replacing y in y=1y=1: −y=1-y=1, hence y=−1y=-1. The vertical asymptote x=−1x=-1 is independent of y and remains unchanged.

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Separate two-branch graphs of 1+1/(x+1) and −1−1/(x+1). Both have vertical asymptote x=−1; horizontal asymptote changes from y=1 to y=−1 under reflection in the x-axis.
Separate two-branch graphs of 1+1/(x+1) and −1−1/(x+1). Both have vertical asymptote x=−1; horizontal asymptote changes from y=1 to y=−1 under reflection in the x-axis.
Separate two-branch graphs of 1+1/(x+1) and −1−1/(x+1). Both have vertical asymptote x=−1; horizontal asymptote changes from y=1 to y=−1 under reflection in the x-axis.
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