How to Write the Equation of Parabola?
In this article, you will learn how to write the equation of a parabola in standard form.
Tutor-style math help
Write the Equation of Parabola: 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
Name the conic first. Circle, ellipse, parabola, and hyperbola have different standard forms and different graph features.
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
Write the Equation of Parabola: 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:
A parabola is a U-shaped line or curve that is defined as a locus of points that the distance to a fixed point (the focus) and a fixed straight line (the directrix) are equal.
Related Topics
Step by Step Guide to Write the Equation of Parabola
- The standard form of Parabola when it opens up or down is \((x- h)^2= 4p(y-k)\), where the vertex is \(h, k\), the focus is \(h,k+p\) and the directrix is \(y=k-p\).
- The standard form of Parabola when it opens right or left is \((y-k)^2= 4p(x-h)\), where the vertex is \(h, k\), the focus is \(h+p,k\) and the directrix is \(x=h-p\).
Equation of Parabola – Example 1:
Write the equation of the parabola with vertex \((1, 4)\) and focus \((0, 6)\)
Solution:
The standard form of Parabola is \((x- h)^2= 4p(y-k)\).
Vertex is \((1, 4)\) then: \(h=1, k=4\)
Focus is \((0, 6)=(h, k+p)\), then: \(k+p=6, k=4 → 4+p=6 →p=2 \)
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Plugin the values in the equation: \((x- 1)^2= 4(2)(y-4)\)
Then: \((x- 1)^2= 8(y-4)\)
Exercises for Writing Equation of Parabola
Write the equation of each Parabola.
- Vertex \((2, 5)\) and focus \((2, 7)\)
- Vertex \((3, 0)\) and focus \((5, 0)\)
- Vertex \((-1, 2)\) and focus \((-1, 0)\)
- Vertex \((0, 3)\) and focus \((-3, 3)\)
- \(\color{blue}{(x-2)^2=8(y-5)}\)
- \(\color{blue}{y^2=8(x-3)}\)
- \(\color{blue}{(x+1)^2=-8(y-2)}\)
- \(\color{blue}{(y-3)^2=-12x}\)
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