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

## Example Questions

### Example Question #225 : Coordinate Geometry

If the -intercept of the line is  and the slope is , which of the following equations best satisfies this condition?

Explanation:

Write the slope-intercept form.

The point given the x-intercept of 6 is .

Substitute the point and the slope into the equation and solve for the y-intercept.

Substitute the y-intercept back to the slope-intercept form to get your equation.

### Example Question #226 : Coordinate Geometry

A vertical parabola on the coordinate plane has vertex  and -intercept

Give its equation.

Insufficient information is given to determine the equation.

Explanation:

The equation of a vertical 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 #1 : How To Find The Equation Of A Curve

A vertical 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 vertical 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 #2 : How To Find The Equation Of A Curve

A vertical parabola on the coordinate plane has -intercept ; its only -intercept is .

Give its equation.

Insufficient information is given to determine the equation.

Explanation:

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

The equation of a vertical 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 #1 : How To Find The Equation Of A Curve

A vertical 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 vertical 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 #4 : How To Find The Equation Of A Curve

Give the equation of the above ellipse.

Explanation:

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

The ellipse has center , horizontal axis of length 8, and vertical axis of length 16. Therefore,

, and .

The equation of the ellipse is

### Example Question #5 : How To Find The Equation Of A Curve

Give the equation of the above ellipse.

Explanation:

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

The ellipse has center , horizontal axis of length 10, and vertical axis of length 6. Therefore,

, and .

The equation of the ellipse is

### Example Question #6 : How To Find The Equation Of A Curve

Give the equation of the above ellipse.

Explanation:

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

The ellipse has center , horizontal axis of length 8, and vertical axis of length 6. Therefore,

, and .

The equation of the ellipse is

### Example Question #7 : How To Find The Equation Of A Curve

The -intercept and the only -intercept of a vertical parabola on the coordinate plane coincide with the -intercept and the -intercept of the line of the equation . Give the equation of the parabola.

Insufficient information is given to determine the equation.

Explanation:

To find the -intercept, that is, the point of intersection with the -axis, of the line of equation , set  and solve for :

The -intercept is .

The -intercept can be found by doing the opposite:

The -intercept is .

The parabola has these intercepts as well. Also, since the vertical parabola has only one -intercept, that point doubles as its vertex as well.

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

,

where  is the vertex. Set :

for some real . To find it, use the -intercept, setting

The parabola has equation , which is rewritten as

### Example Question #8 : How To Find The Equation Of A Curve

An ellipse on the coordinate plane has as its center the point . It passes through the points  and . Give its equation.

Insufficient information is given to determine the equation.

Explanation:

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

The center is , so  and .

To find , note that one endpoint of the horizontal axis is given by the point with the same -coordinate through which it passes, namely, . Half the length of this axis, which is , is the difference of the -coordinates, so . Similarly, to find , note that one endpoint of the vertical axis is given by the point with the same -coordinate through which it passes, namely, . Half the length of this axis, which is , is the difference of the -coordinates, so .

The equation is

or

.