# SSAT Upper Level Math : How to find the equation of a curve

## Example Questions

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### Example Question #231 : Coordinate Geometry

An ellipse passes through points

Give its equation.

Explanation:

The equation of the ellipse with center , horizontal axis of length , and vertical axis of length  is

and  are the endpoints of a horizontal line segment with midpoint

, or

and length .

and  are the endpoints of a vertical line segment with midpoint

, or

and length

Because their midpoints coincide, these are the endpoints of the horizontal axis and vertical axis, respectively, of the ellipse, and the common midpoint  is the center.

Therefore,

and ;

and ; consequently  and .

The equation of the ellipse is

, or

### Example Question #21 : X And Y Intercept

A horizontal parabola on the coordinate plane   as its only -intercept; its -intercept is .

Give its equation.

Insufficient information is given to determine the equation.

Explanation:

If a horizontal parabola has only one -intercept, which here is , that point doubles as its vertex as well.

The equation of a horizontal parabola, in vertex form, is

,

where  is the vertex. Set :

To find , use the -intercept, setting :

The equation, in vertex form, is . In standard form:

### Example Question #28 : X And Y Intercept

A horizontal parabola on the coordinate plane has vertex ; one of its -intercepts is .

Give its equation.

Insufficient information is given to determine the equation.

Explanation:

The equation of a horizontal parabola, in vertex form, is

,

where  is the vertex. Set :

To find , use the known -intercept, setting :

The equation, in vertex form, is ; in standard form:

### Example Question #29 : X And Y Intercept

A vertical parabola on the coordinate plane has -intercepts  and , and passes through .

Give its equation.

Insufficient information is given to determine the equation.

Explanation:

A horizontal parabola which passes through   and  has as its equation

.

To find , substitute the coordinates of the third point, setting :

The equation is therefore , which is, in standard form:

### Example Question #30 : X And Y Intercept

A vertical parabola on the coordinate plane has -intercepts  and , and passes through .

Give its equation.

Explanation:

A vertical parabola which passes through  and  has as its equation

To find , substitute the coordinates of the third point, setting :

The equation is ; expand to put it in standard form:

### Example Question #11 : How To Find The Equation Of A Curve

A horizontal parabola on the coordinate plane has -intercept ; one of its -intercepts is .

Give its equation.

Insufficient information is given to determine the equation.

Insufficient information is given to determine the equation.

Explanation:

The equation of a horizontal parabola, in standard form, is

for some real

is the -coordinate of the -intercept, so , and the equation is

Set :

However, no other information is given, so the values of  and  cannot be determined for certain. The correct response is that insufficient information is given.

### Example Question #31 : X And Y Intercept

A vertical parabola on the coordinate plane includes points  and

Give its equation.

Explanation:

The standard form of the equation of a vertical parabola is

If the values of  and  from each ordered pair are substituted in succession, three equations in three variables are formed:

The system

can be solved through the elimination method.

First, multiply the second equation by  and add to the third:

Next, multiply the second equation by  and add to the first:

Which can be divided by 3 on both sides to yield

Now solve the two-by-two system

by substitution:

Back-solve:

Back-solve again:

The equation of the parabola is therefore

.

### Example Question #33 : X And Y Intercept

A vertical parabola on the coordinate plane shares one -intercept with the line of the equation , and the other with the line of the equation . It also passes through . Give the equation of the parabola.

The correct answer is not among the other responses.

Explanation:

First, find the -intercepts—the points of intersection with the -axis—of the lines by substituting 0 for  in both equations.

is the -intercept of this line.

is the -intercept of this line.

The parabola has -intercepts at  and , so its equation can be expressed as

for some real . To find it, substitute using the coordinates of the third point, setting :

.

The equation is , which, in standard form, can be rewritten as:

### Example Question #34 : X And Y Intercept

A horizontal parabola on the coordinate plane includes points  , and

Give its equation.

Explanation:

The standard form of the equation of a horizontal parabola is

If the values of  and  from each ordered pair are substituted in succession, three equations in three variables are formed:

The three-by-three linear system

can be solved by way of the elimination method.

can be found first, by multiplying the first equation by  and add it to the second:

Substitute 5 for  in the last two equations to form a two-by-two linear system:

The system

can be solved by way of the substitution method;

Substitute 2 for  in the top equation:

The equation is .

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