Plotting these points, we have two curves, one in the first quadrant and one in the third quadrant. These two are referred to as one curve having two branches.

Example 3. Trace the curve 2+2y216.

If x = 0, y =±√8; if y = 0, x = ±4. In the equation x2 + 2 y2 = 16, the value of x does not exceed 4, and the value of y does not exceed +√8.

For x=0, 1, ±2, ±3, ±3.5, ±√14, ±4, we find,

y = =±√8, ±√7.5, ±√6, ±√3.5, ±√1.87, ±1, 0.

In plotting these points it will be sufficient to get the values of ±√8, ±√7.5, ±√6, etc., correct to two decimal figures.

Example 4. Trace the curve x2+ y2 = 12.

For x=0, 11, 12, ±2.5, ±3, ±√12,

we find,

y=±√12, ±√11, ±√8, ±√5.75, ±√3, 0.

Plotting the points, we get a circle whose radius is ±√12.

we find by subtracting (1) from (2), member by member,

y2 = 4,
y= ±2.

Substituting in either equation ±2 for y, we get

x=±√8.

These values of x and y are the coördinates of the four points of intersection of the two curves. This is as it should be, for the coördinates of every point on a curve satisfy the equation of the curve, and consequently if two curves have points in common, the coördinates of these points must satisfy both equations of the curves.

4. y=(x-2)2, y = (x-3)2, y=(x+3)2.

5. x2+ y2 = 25, x2+ y2 = 16.

6. x2— y2=7, x2 — y2 = 12, x2 — y2 = 1.

7. y2=8x, y2 = 16x, y2 = 4x.

8. 2x2+3y2 = 32, 4 x2 + 9 y2 = 36.

9. y=x(6−x), y=x(4−x), y=x(1−x).

HINT. In the last one take x = .2, .5, ±.7, ±1, ±2, etc.

10. Solve x2-x+1=0. Trace the graph, y=x2−x+1. How is it situated with respect to the x-axis?

11. The number of feet a body falls under the influence of gravity in t seconds is given by the formula S = 16 ť2, where S denotes the number of feet. Trace the graph of S= 16 t2.

From equation (2), we find, y = 5 — x.

Substitute this value of y in equation (1), and we have,

x2 + 2 (5 − x)2 = 17.

(3)

Solving equation (3), we find, x = 3 or 11.

Substituting this value of x in equation (2), y = 2, or .

Systems of equations in two variables where one equation is quadratic and the other linear may be all solved like equation (1) above.