Hyperbola in Standard Form and Vertices, Co– Vertices, Foci, and Asymptotes of a Hyperbola
| \(\frac{(y-k)^2}{a^2}-\frac{(x-h)^2}{b^2}=1\) | \(\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1\) |
| Center: \((h, k)\) Foci: \((h, k ± c)\) Vertices: \((h, k ± a)\) Transverse axis: \(x=h\) (Parallel to \(y\)-axis) Asymptotes: \(y-k=±\frac{a}{b}(x-h)\) |
Center: \((h, k)\) Foci: \((h ± c, k)\) Vertices: \((h ± a, k)\) Transverse axis: \((y=k)\) (Parallel to \(x\) -axis) Asymptotes: \(y-k=±\frac{b}{a}(x-h)\) |
Hyperbola in Standard Form and Vertices, Co– Vertices, Foci, and Asymptotes of a Hyperbola – Example 1:
Solution:
Tutor-style math help
Hyperbola in Standard Form and Vertices, Co– Vertices, Foci, and Asymptotes of a Hyperbola: what to notice and how to work it
Conics skill
Conic sections are graph shapes with standard forms. The equation tells you the center or vertex, then the key distances shape the graph.
What to notice first
For a hyperbola, identify the center, transverse direction, vertices, and asymptotes before drawing branches.
Common student mistake
Do not read signs backward in \((x-h)\) and \((y-k)\). The center or vertex uses \(h\) and \(k\), not the visible sign alone.
Key formulas and cues
\((x-h)^2+(y-k)^2=r^2\)
\(\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1\)
\((y-k)=a(x-h)^2\)
\(\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1\)
A reliable path
- Match the formIdentify the conic by its equation pattern.
- Read featuresFind the center, vertex, radius, axes, foci, or asymptotes.
- Sketch from anchorsPlot key points first, then draw the curve.
Worked examples
Circle center and radius
Example: \((x-4)^2+(y+1)^2=25\)
- Compare to circle standard form.
- The center is (4, -1).
- The radius is the square root of 25.
Answer: Center (4, -1), radius 5
Parabola direction
Example: \((x-2)^2=8(y+3)\)
- The x part is squared.
- The parabola opens up or down.
- The positive coefficient means it opens up.
Answer: Opens up
Try one before moving on
Try: Find the center of \((x+3)^2+(y-2)^2=16\).
Answer: (-3, 2).
Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.
x
Hyperbola in Standard Form and Vertices, Co– Vertices, Foci, and Asymptotes of a Hyperbola: pop-up practice
Answer these quick questions, then use the feedback to decide which part of the lesson to review.
Choose an answer to begin.
1. \((x-1)^2+(y+2)^2=9\) has center:
2. A hyperbola usually has:
3. An ellipse has:
To rewrite in standard form, first add \(44\) to both sides: \(x^2+y^2+8x-4y=44\)
Group \(x\) -variables and \(y\) -variables together: \((x^2+8x)+(y^2-4y)=44\)
Convert \(x\) and \(y\) to square form: \((x^2+8x+16)+(y^2-4y+4)=44+16+4 → (x+4)^2+(y-2)^2=64\)
Divide by \(64\): \(\frac{(x+4)^2}{64}-\frac{(y-2)^2}{64}=1\)
Then: \((h, k)=(-4, 2), a=8, b=8,\) and center is \((-4, 2)\)
Foci: \((-4, 2+c), (-4,2-c)\)
Compute \(c: c=\sqrt{8^2+8^2}=8\sqrt{2}\) then: \((-4, 2+8\sqrt{2}), (-4, 2-8\sqrt{2})\)
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