Algebra Know-It-ALL

Part Three 533

of the other real solution will become more positive to the same extent. The y-values of both
solutions will remain at 0; the points will stay on the x axis. Figure 30-5 shows an example.
On both axes, each increment represents 1 unit. The stand-alone constants in the equations
will change. The negative constant in the first equation will become more negative, and the
positive constant in the second equation will become more positive to the same extent.

Question 28-5

Let’s modify the system presented in Question 28-4 and graphed in Fig. 30-5. Suppose that
we move the upward-opening parabola even further straight down, but leave the downward-
opening parabola in the same place. What will happen to the solution points? How will the
equations in the system change?

Answer 28-5

The intersection points will move even farther from each other. The negative x-value of one
real solution will become more negative, and the positive x-value of the other real solution will
become more positive to the same extent. The y-values of both solutions will become nega-
tive to an equal extent. The solution points will move off the x axis into the third and fourth
quadrants of the coordinate plane. This assumes that we move the upward-opening parabola
exactly in the negative-y direction. Figure 30-6 shows an example. On both axes, each incre-
ment represents 1 unit. The negative constant in the first equation will become more negative,
and the positive constant in the second equation will stay the same.

Question 28-6

Consider the system of equations we solved in Answer 27-6:

y= (x+ 1)^2

x

y

Solution Solution

Figure 30-5 Illustration for Answer 28-4. The first function

is graphed as a solid curve; the second function is

graphed as a dashed curve. Real-number solutions

appear as points where the curves intersect. On both

axes, each increment is 1 unit.