Equation of parabola with vertex and directrix

Q: How do you find the equation of a parabola given the vertex?

We can use the vertex form to find a parabolas equation. The idea is to use the coordinates of its vertex (maximum point, or minimum point) to write its equation in the form y=a(x−h)2+k (assuming we can read the coordinates (h,k) from the graph) and then to find the value of the coefficient a.

This calculator will find either the equation of the parabola from the given parameters or the vertex, focus, directrix, axis of symmetry, latus rectum, length of the latus rectum, focal parameter, focal length (distance), eccentricity, x-intercepts, y-intercepts, domain, and range of the entered parabola. Also, it will graph the parabola. Steps are available.

Table of Contents Show

How do you find the equation of a parabola with the Directrix?
How do you find the equation of a parabola given the vertex?
Why is a parabola formula 4p?

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Your Input

Find the vertex, focus, directrix, axis of symmetry, latus rectum, length of the latus rectum, focal parameter, focal length, eccentricity, x-intercepts, y-intercepts, domain, and range of the parabola $$$y = \left(x - 2\right)^{2} + 5$$$.

Solution

The equation of a parabola is $$$y = \frac{1}{4 \left(f - k\right)} \left(x - h\right)^{2} + k$$$, where $$$\left(h, k\right)$$$ is the vertex and $$$\left(h, f\right)$$$ is the focus.

Our parabola in this form is $$$y = \frac{1}{4 \left(\frac{21}{4} - 5\right)} \left(x - 2\right)^{2} + 5$$$.

Thus, $$$h = 2$$$, $$$k = 5$$$, $$$f = \frac{21}{4}$$$.

The standard form is $$$y = x^{2} - 4 x + 9$$$.

The general form is $$$x^{2} - 4 x - y + 9 = 0$$$.

The vertex form is $$$y = \left(x - 2\right)^{2} + 5$$$.

The directrix is $$$y = d$$$.

To find $$$d$$$, use the fact that the distance from the focus to the vertex is the same as the distance from the vertex to the directrix: $$$5 - \frac{21}{4} = d - 5$$$.

Thus, the directrix is $$$y = \frac{19}{4}$$$.

The axis of symmetry is the line perpendicular to the directrix that passes through the vertex and the focus: $$$x = 2$$$.

The focal length is the distance between the focus and the vertex: $$$\frac{1}{4}$$$.

The focal parameter is the distance between the focus and the directrix: $$$\frac{1}{2}$$$.

The latus rectum is parallel to the directrix and passes through the focus: $$$y = \frac{21}{4}$$$.

The endpoints of the latus rectum can be found by solving the system $$$\begin{cases} x^{2} - 4 x - y + 9 = 0 \\ y = \frac{21}{4} \end{cases}$$$ (for steps, see system of equations calculator).

The endpoints of the latus rectum are $$$\left(\frac{3}{2}, \frac{21}{4}\right)$$$, $$$\left(\frac{5}{2}, \frac{21}{4}\right)$$$.

The length of the latus rectum is four times the distance between the vertex and the focus: $$$1$$$.

The eccentricity of a parabola is always $$$1$$$.

The x-intercepts can be found by setting $$$y = 0$$$ in the equation and solving for $$$x$$$ (for steps, see intercepts calculator).

Since there are no real solutions, there are no x-intercepts.

The y-intercepts can be found by setting $$$x = 0$$$ in the equation and solving for $$$y$$$: (for steps, see intercepts calculator).

y-intercept: $$$\left(0, 9\right)$$$.

Answer

Standard form: $$$y = x^{2} - 4 x + 9$$$A.

General form: $$$x^{2} - 4 x - y + 9 = 0$$$A.

Vertex form: $$$y = \left(x - 2\right)^{2} + 5$$$A.

Focus-directrix form: $$$\left(x - 2\right)^{2} + \left(y - \frac{21}{4}\right)^{2} = \left(y - \frac{19}{4}\right)^{2}$$$A.

Graph: see the graphing calculator.

Vertex: $$$\left(2, 5\right)$$$A.

Focus: $$$\left(2, \frac{21}{4}\right) = \left(2, 5.25\right)$$$A.

Directrix: $$$y = \frac{19}{4} = 4.75$$$A.

Axis of symmetry: $$$x = 2$$$A.

Latus rectum: $$$y = \frac{21}{4} = 5.25$$$A.

Endpoints of the latus rectum: $$$\left(\frac{3}{2}, \frac{21}{4}\right) = \left(1.5, 5.25\right)$$$, $$$\left(\frac{5}{2}, \frac{21}{4}\right) = \left(2.5, 5.25\right)$$$A.

Length of the latus rectum: $$$1$$$A.

Focal parameter: $$$\frac{1}{2} = 0.5$$$A.

Focal length: $$$\frac{1}{4} = 0.25$$$A.

Eccentricity: $$$1$$$A.

x-intercepts: no x-intercepts.

y-intercept: $$$\left(0, 9\right)$$$A.

Domain: $$$\left(-\infty, \infty\right)$$$A.

Range: $$$\left[5, \infty\right)$$$A.

How do you find the equation of a parabola with the Directrix?

How Do I Find Directrix of a Parabola? The directrix of a parabola can be found, by knowing the axis of the parabola, and the vertex of the parabola. For an equation of the parabola in standard form y² = 4ax, with focus at (a, 0), axis as the x-axis, the equation of the directrix of this parabola is x + a = 0 .

How do you find the equation of a parabola given the vertex?

We can use the vertex form to find a parabola's equation. The idea is to use the coordinates of its vertex (maximum point, or minimum point) to write its equation in the form y=a(x−h)2+k (assuming we can read the coordinates (h,k) from the graph) and then to find the value of the coefficient a.

Why is a parabola formula 4p?

Finding p gives us the distance between the vertex and the focus and the vertex and the directrix. It's a twofer. The value 4p is attached to the unsquared part of the equation, so divide that by 4 to get to p.