What is the difference between standard and vertex form of a parabola?

Standard form uses squared terms on one side, while vertex form shows the vertex clearly and is easier for graphing.

How do you find the equation of a parabola given 3 points?

Use a system of equations by substituting the three points into the general form y=ax^2 + bx + c.

What is the equation of the directrix?

For (x-h)^2, directrix is y = k - a. For horizontal parabola, it’s x = h - a.

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Equation of a Parabola

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Equation of a Parabola

The equation of a parabola represents a U-shaped curve formed by the graph of a quadratic function. It’s widely used in mathematics, physics, and engineering to model real-life situations such as projectile motion, satellite dishes, and optics. Understanding how to derive, graph, and transform the equation of a parabola is crucial for both high school and competitive exams.

1.0What is a Parabola?

A parabola is a set of points that are equidistant from a fixed point (focus) and a fixed line (directrix). It opens either up, down, left, or right, depending on the orientation of the axis of symmetry.

2.0Equation of a Parabola in Standard Form

The standard form of a parabola depends on the axis of symmetry:

Vertical Parabola (opens up/down): $(x - h)^{2} = 4 a (y - k)$
Horizontal Parabola (opens left/right): $(y - k)^{2} = 4 a (x - h)$

Where:

(h, k) is the vertex,
aa is the distance from the vertex to the focus,
The directrix is a fixed line located opposite the focus.

3.0Equation of a Parabola in Vertex Form

For a vertical parabola: $y = a (x - h)^{2} + k$

Here:

aa controls the direction and width of the parabola,
(h, k) is the vertex.

Determining the Equation of a Parabola:

To determine the equation, you need:

The vertex of the parabola,
A point on the curve (or focus and directrix),
The direction it opens.

4.0Parametric Equation of a Parabola

For the parabola $(x - h)^{2} = 4 a (y - k)$ , the parametric equations are: $x = h + 2 a t, y = k_{a} t^{2}$

Where t is the parameter.

5.0Graphing Parabola Equations

Graphing parabola equations involves:

Identifying the vertex.
Determining the direction the parabola opens.
Plotting the focus and directrix.
Drawing the symmetric curve.

Use a graphing calculator or plotting software for accuracy, especially when the parabola is rotated or translated.

Also Explore: Graphical Method Linear Programming

6.0Parabola Equation Examples

Example 1: Find the equation of a parabola with vertex at (0, 0) and focus at (0, 3).

Solution:

Since it opens upward, use: $(x - h)^{2} = 4 a (y - k)$
a = 3, h = 0, k = 0

$\Rightarrow x^{2} = 12 y$

Example 2: Find the equation of the directrix of the parabola $y = 2 (x + 1)^{2} + 3$

Solution:

This is in vertex form, vertex is (1, 3), a = 2

Directrix: $y = k - \frac{1}{4 a} = 3 - \frac{1}{8} = \frac{23}{8}$

Example 3: Find the equation of the parabola whose focus is at (3, 0) and directrix is the line x = -3. Also find its vertex.

Solution:

The vertex lies midway between focus and directrix: $V = (\frac{3 + ( - 3 )}{2}, 0) = (0, 0)$
Distance from vertex to focus = 3
So, a = 3, opening rightwards.
Standard form: $y^{2} = 4 a x \Rightarrow y^{2} = 12 x$

Answer: $y^{2} = 12 x$ ; Vertex: (0, 0)

Example 4: Find the equation of the tangent to the parabola $y^{2} = 4 x$ that makes an angle of $4 5^{0}$ with the x-axis.

Solution:

Slope of line = $m = t an (4 5^{0}) = 1$
Equation of tangent in slope form:

$y = m x + \frac{a}{m}$ , where a = 1

$\Rightarrow y = x + \frac{1}{1} = x + 1$

Answer: y = x + 1

Example 5: A chord of the parabola $y^{2} = 4 a x$ passes through the focus and is perpendicular to the axis. Find its length.

Solution:

Focus: (a, 0)
Line perpendicular to x-axis through focus: x = a
Sub into parabola: $y^{2} = 4 a^{2} \Rightarrow y = \pm 2 a$
Points: (a, 2a), (a, -2a)
Length $= 2 \times 2 a = 4 a$

Answer: 4a

Example 6: Find the coordinates of the foot of perpendicular from the focus of the parabola $y^{2} = 8 x$ on the line y = x + 2.

Solution:

Focus of $y^{2} = 8 x$ : (2, 0)
Perpendicular from (2, 0) to y = x + 2:

Slope of given line = 1 → slope of perpendicular = -1

Equation of perpendicular:

Passes through (2, 0), slope -1 $\Rightarrow y - 0 = - 1 (x - 2) \Rightarrow y = - x + 2$

Solve with y = x + 2 and y = -x + 2 $\Rightarrow x + 2 = - x + 2 \Rightarrow 2 x = 0 \Rightarrow x = 0, y = 2$

Answer: (0, 2)

Example 7: Find the locus of a point which moves such that its distance from the point (0, 3) is equal to its distance from the line y = -3.

Solution:
Let point be (x, y)

Distance from (0, 3): $x^{2} + (y - 3)^{2}$
Distance from line $y = - 3 y = - 3 : ∣ y + 3∣∣ y + 3∣$

Equating: $x^{2} + (y - 3)^{2} = ∣ y + 3∣$

Squaring:

$x^{2} + (y - 3)^{2} = (y + 3)^{2}$

$\Rightarrow x^{2} + y^{2} - 6 y + 9 = y^{2} + 6 y + 9$

$\Rightarrow x^{2} = 12 y$

Answer: $x^{2} = 12 y$ a vertical parabola

7.0Practice Questions on Equation of Parabola

Convert $x = - 2 y^{2} + 4 y - 1$ to standard form and identify vertex and axis.
Find the parametric form of the parabola $x^{2} = 8 y$ .
Graph the parabola $y = - 3 (x + 1)^{2} + 2$ .
Determine the equation of the directrix for $x^{2} = - 16 y$ .

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Equation of a Parabola

The equation of a parabola represents a U-shaped curve formed by the graph of a quadratic function. It’s widely used in mathematics, physics, and engineering to model real-life situations such as projectile motion, satellite dishes, and optics. Understanding how to derive, graph, and transform the equation of a parabola is crucial for both high school and competitive exams.

1.0What is a Parabola?

A parabola is a set of points that are equidistant from a fixed point (focus) and a fixed line (directrix). It opens either up, down, left, or right, depending on the orientation of the axis of symmetry.

2.0Equation of a Parabola in Standard Form

The standard form of a parabola depends on the axis of symmetry:

Vertical Parabola (opens up/down): $(x - h)^{2} = 4 a (y - k)$
Horizontal Parabola (opens left/right): $(y - k)^{2} = 4 a (x - h)$

Where:

(h, k) is the vertex,
aa is the distance from the vertex to the focus,
The directrix is a fixed line located opposite the focus.

3.0Equation of a Parabola in Vertex Form

For a vertical parabola: $y = a (x - h)^{2} + k$

Here:

aa controls the direction and width of the parabola,
(h, k) is the vertex.

Determining the Equation of a Parabola:

To determine the equation, you need:

The vertex of the parabola,
A point on the curve (or focus and directrix),
The direction it opens.

4.0Parametric Equation of a Parabola

For the parabola $(x - h)^{2} = 4 a (y - k)$ , the parametric equations are: $x = h + 2 a t, y = k_{a} t^{2}$

Where t is the parameter.

5.0Graphing Parabola Equations

Graphing parabola equations involves:

Identifying the vertex.
Determining the direction the parabola opens.
Plotting the focus and directrix.
Drawing the symmetric curve.

Use a graphing calculator or plotting software for accuracy, especially when the parabola is rotated or translated.

Also Explore: Graphical Method Linear Programming

6.0Parabola Equation Examples

Example 1: Find the equation of a parabola with vertex at (0, 0) and focus at (0, 3).

Solution:

Since it opens upward, use: $(x - h)^{2} = 4 a (y - k)$
a = 3, h = 0, k = 0

$\Rightarrow x^{2} = 12 y$

Example 2: Find the equation of the directrix of the parabola $y = 2 (x + 1)^{2} + 3$

Solution:

This is in vertex form, vertex is (1, 3), a = 2

Directrix: $y = k - \frac{1}{4 a} = 3 - \frac{1}{8} = \frac{23}{8}$

Example 3: Find the equation of the parabola whose focus is at (3, 0) and directrix is the line x = -3. Also find its vertex.

Solution:

The vertex lies midway between focus and directrix: $V = (\frac{3 + ( - 3 )}{2}, 0) = (0, 0)$
Distance from vertex to focus = 3
So, a = 3, opening rightwards.
Standard form: $y^{2} = 4 a x \Rightarrow y^{2} = 12 x$

Answer: $y^{2} = 12 x$ ; Vertex: (0, 0)

Example 4: Find the equation of the tangent to the parabola $y^{2} = 4 x$ that makes an angle of $4 5^{0}$ with the x-axis.

Solution:

Slope of line = $m = t an (4 5^{0}) = 1$
Equation of tangent in slope form:

$y = m x + \frac{a}{m}$ , where a = 1

$\Rightarrow y = x + \frac{1}{1} = x + 1$

Answer: y = x + 1

Example 5: A chord of the parabola $y^{2} = 4 a x$ passes through the focus and is perpendicular to the axis. Find its length.

Solution:

Focus: (a, 0)
Line perpendicular to x-axis through focus: x = a
Sub into parabola: $y^{2} = 4 a^{2} \Rightarrow y = \pm 2 a$
Points: (a, 2a), (a, -2a)
Length $= 2 \times 2 a = 4 a$

Answer: 4a

Example 6: Find the coordinates of the foot of perpendicular from the focus of the parabola $y^{2} = 8 x$ on the line y = x + 2.

Solution:

Focus of $y^{2} = 8 x$ : (2, 0)
Perpendicular from (2, 0) to y = x + 2:

Slope of given line = 1 → slope of perpendicular = -1

Equation of perpendicular:

Passes through (2, 0), slope -1 $\Rightarrow y - 0 = - 1 (x - 2) \Rightarrow y = - x + 2$

Solve with y = x + 2 and y = -x + 2 $\Rightarrow x + 2 = - x + 2 \Rightarrow 2 x = 0 \Rightarrow x = 0, y = 2$

Answer: (0, 2)

Example 7: Find the locus of a point which moves such that its distance from the point (0, 3) is equal to its distance from the line y = -3.

Solution:
Let point be (x, y)

Distance from (0, 3): $x^{2} + (y - 3)^{2}$
Distance from line $y = - 3 y = - 3 : ∣ y + 3∣∣ y + 3∣$

Equating: $x^{2} + (y - 3)^{2} = ∣ y + 3∣$

Squaring:

$x^{2} + (y - 3)^{2} = (y + 3)^{2}$

$\Rightarrow x^{2} + y^{2} - 6 y + 9 = y^{2} + 6 y + 9$

$\Rightarrow x^{2} = 12 y$

Answer: $x^{2} = 12 y$ a vertical parabola

7.0Practice Questions on Equation of Parabola

Convert $x = - 2 y^{2} + 4 y - 1$ to standard form and identify vertex and axis.
Find the parametric form of the parabola $x^{2} = 8 y$ .
Graph the parabola $y = - 3 (x + 1)^{2} + 2$ .
Determine the equation of the directrix for $x^{2} = - 16 y$ .

Frequently Asked Questions

What is the difference between standard and vertex form of a parabola?

How do you find the equation of a parabola given 3 points?

What is the equation of the directrix?

Frequently Asked Questions

What is the difference between standard and vertex form of a parabola?

How do you find the equation of a parabola given 3 points?

What is the equation of the directrix?

Join ALLEN!

Equation of a Parabola

1.0What is a Parabola?

2.0Equation of a Parabola in Standard Form

3.0Equation of a Parabola in Vertex Form

4.0Parametric Equation of a Parabola

5.0Graphing Parabola Equations

6.0Parabola Equation Examples

7.0Practice Questions on Equation of Parabola

Table of Contents

Join ALLEN!

Equation of a Parabola

1.0What is a Parabola?

2.0Equation of a Parabola in Standard Form

3.0Equation of a Parabola in Vertex Form

4.0Parametric Equation of a Parabola

5.0Graphing Parabola Equations

6.0Parabola Equation Examples

7.0Practice Questions on Equation of Parabola

Table of Contents