Porisma

Worked solution

Exact intercepts of a translated vertical hyperbola

Sketch (y+3)24−(x−2)29=1\dfrac{(y+3)^2}{4}-\dfrac{(x-2)^2}{9}=1 and write down the coordinates of the intercepts, vertices and the equations of all asymptotes and lines of symmetry.

CJC · Lecture · Example 7

Centre and vertices

[y−(−3)]222−(x−2)232=1\dfrac{[y-(-3)]^2}{2^2}-\dfrac{(x-2)^2}{3^2}=1, so centre (2,−3)(2,-3), vertical vertex distance 22 and vertices (2,−1),(2,−5)(2,-1),(2,-5). The curve opens up and down.

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Exact axial intercepts

At y=0y=0, 9/4−(x−2)2/9=19/4-(x-2)^2/9=1, so (x−2)2=45/4(x-2)^2=45/4. The xx-intercepts are (2±35/2,0)(2\pm3\sqrt5/2,0). At x=0x=0, (y+3)2/4−4/9=1(y+3)^2/4-4/9=1, so (y+3)2=52/9(y+3)^2=52/9. The yy-intercepts are (0,−3±213/3)(0,-3\pm2\sqrt{13}/3). Keep the radicals exact as printed; the Teacher copy supplies these labels on the Student sketch.

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Asymptotes, symmetry and sketch

Set the standard equation’s RHS to zero: (y+3)2/4=(x−2)2/9(y+3)^2/4=(x-2)^2/9. Thus (y+3)2=49(x−2)2(y+3)^2=\tfrac49(x-2)^2 and y+3=±23(x−2)y+3=\pm\tfrac23(x-2). The two asymptotes are y=−23x−53y=-\tfrac23x-\tfrac53 and y=23x−133y=\tfrac23x-\tfrac{13}{3}. The symmetry lines are x=2,y=−3x=2,y=-3. Draw both smooth branches through the exact intercepts and vertices, approaching the two asymptotes. The centre is not a point on either branch.

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Compare your sketch

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Vertical hyperbola centred at (2,−3), vertices (2,−1),(2,−5), x-intercepts (2±3√5/2,0), y-intercepts (0,−3±2√13/3), asymptotes y=(2/3)x−13/3 and y=−(2/3)x−5/3, symmetry x=2,y=−3.
Vertical hyperbola centred at (2,−3), vertices (2,−1),(2,−5), x-intercepts (2±3√5/2,0), y-intercepts (0,−3±2√13/3), asymptotes y=(2/3)x−13/3 and y=−(2/3)x−5/3, symmetry x=2,y=−3.
Vertical hyperbola centred at (2,−3), vertices (2,−1),(2,−5), x-intercepts (2±3√5/2,0), y-intercepts (0,−3±2√13/3), asymptotes y=(2/3)x−13/3 and y=−(2/3)x−5/3, symmetry x=2,y=−3.

Common mistakes

The vertical vertices are not axial intercepts unless they lie on an axis. Retain both ± roots in each intercept calculation.

Solution notes

Student/Teacher PDF 23, printed 135. All printed intercepts, vertices, asymptote derivation and symmetry lines retained. Teacher coordinate labels integrated directly into the sketch and working; no marks printed.

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