Find parabola given the point $$$\left(0, 0\right)$$$, the point $$$\left(20, 35\right)$$$, the point $$$\left(80, 0\right)$$$

Find the equation, vertex, focus, directrix, axis of symmetry, latus rectum, length of the latus rectum (focal width), focal parameter, focal length, eccentricity, x-intercepts, y-intercepts, domain, and range of the parabola found from the given data: the point $$$\left(0, 0\right)$$$, the point $$$\left(20, 35\right)$$$, the point $$$\left(80, 0\right)$$$.

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.

Since the point $$$\left(0, 0\right)$$$ lies on the parabola, then $$$0 = \frac{1}{4 \left(f - k\right)} \left(0 - h\right)^{2} + k$$$.

Since the point $$$\left(20, 35\right)$$$ lies on the parabola, then $$$35 = \frac{1}{4 \left(f - k\right)} \left(20 - h\right)^{2} + k$$$.

Since the point $$$\left(80, 0\right)$$$ lies on the parabola, then $$$0 = \frac{1}{4 \left(f - k\right)} \left(80 - h\right)^{2} + k$$$.

Solving the system $$$\begin{cases} 0 = \frac{h^{2}}{4 f - 4 k} + k \\ 35 = k + \frac{\left(20 - h\right)^{2}}{4 f - 4 k} \\ 0 = k + \frac{\left(80 - h\right)^{2}}{4 f - 4 k} \end{cases}$$$, we get that $$$h = 40$$$, $$$k = \frac{140}{3}$$$, $$$f = \frac{800}{21}$$$ (for steps, see system of equations calculator).

The standard form is $$$y = - \frac{7 x^{2}}{240} + \frac{7 x}{3}$$$.

The general form is $$$- 7 x^{2} + 560 x - 240 y = 0$$$.

The vertex form is $$$y = - \frac{7 \left(x - 40\right)^{2}}{240} + \frac{140}{3}$$$.

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: $$$\frac{140}{3} - \frac{800}{21} = d - \frac{140}{3}$$$.

Thus, the directrix is $$$y = \frac{1160}{21}$$$.

The axis of symmetry is the line perpendicular to the directrix that passes through the vertex and the focus: $$$x = 40$$$.

The focal length is the distance between the focus and the vertex: $$$\frac{60}{7}$$$.

The focal parameter is the distance between the focus and the directrix: $$$\frac{120}{7}$$$.

The latus rectum is parallel to the directrix and passes through the focus: $$$y = \frac{800}{21}$$$.

The endpoints of the latus rectum can be found by solving the system $$$\begin{cases} - 7 x^{2} + 560 x - 240 y = 0 \\ y = \frac{800}{21} \end{cases}$$$ (for steps, see system of equations calculator).

The endpoints of the latus rectum are $$$\left(\frac{160}{7}, \frac{800}{21}\right)$$$, $$$\left(\frac{400}{7}, \frac{800}{21}\right)$$$.

The length of the latus rectum (focal width) is four times the distance between the vertex and the focus: $$$\frac{240}{7}$$$.

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).

x-intercepts: $$$\left(0, 0\right)$$$, $$$\left(80, 0\right)$$$

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

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

Standard form/equation: $$$y = - \frac{7 x^{2}}{240} + \frac{7 x}{3}$$$A.

General form/equation: $$$- 7 x^{2} + 560 x - 240 y = 0$$$A.

Vertex form/equation: $$$y = - \frac{7 \left(x - 40\right)^{2}}{240} + \frac{140}{3}$$$A.

Focus-directrix form/equation: $$$\left(x - 40\right)^{2} + \left(y - \frac{800}{21}\right)^{2} = \left(y - \frac{1160}{21}\right)^{2}$$$A.

Intercept form/equation: $$$y = - \frac{7 x \left(x - 80\right)}{240}$$$A.

Graph: see the graphing calculator.

Vertex: $$$\left(40, \frac{140}{3}\right)\approx \left(40, 46.666666666666667\right)$$$A.

Focus: $$$\left(40, \frac{800}{21}\right)\approx \left(40, 38.095238095238095\right)$$$A.

Directrix: $$$y = \frac{1160}{21}\approx 55.238095238095238$$$A.

Axis of symmetry: $$$x = 40$$$A.

Latus rectum: $$$y = \frac{800}{21}\approx 38.095238095238095$$$A.

Endpoints of the latus rectum: $$$\left(\frac{160}{7}, \frac{800}{21}\right)\approx \left(22.857142857142857, 38.095238095238095\right)$$$, $$$\left(\frac{400}{7}, \frac{800}{21}\right)\approx \left(57.142857142857143, 38.095238095238095\right)$$$A.

Length of the latus rectum (focal width): $$$\frac{240}{7}\approx 34.285714285714286$$$A.

Focal parameter: $$$\frac{120}{7}\approx 17.142857142857143$$$A.

Focal length: $$$\frac{60}{7}\approx 8.571428571428571$$$A.

Eccentricity: $$$1$$$A.

x-intercepts: $$$\left(0, 0\right)$$$, $$$\left(80, 0\right)$$$A.

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

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

Range: $$$\left(-\infty, \frac{140}{3}\right]\approx \left(-\infty, 46.666666666666667\right]$$$A.