Algebra Know-It-ALL

Part Three 535

Question 28-7

Let’s modify the system of equations stated in Question 28-6. Suppose we multiply the right
side of the second equation by −1, producing this two-by-two system:

y= (x+ 1)^2

and

y=−(x− 1)^2

How will this affect the graph of the second equation, shown by the dashed curve? How will
it affect the real solution set?

Answer 28-7

If we multiply the right side of this equation by −1, we multiply all values of the function by
−1. This inverts the entire graph of the function with respect to the x axis. Figure 30-8 shows
the result. On the x axis, each increment is 1/2 unit. On the y axis, each increment is 2 units.
We can see that the system has no real solutions because the curves don’t intersect. The real
solution set is empty.

Question 28-8

Let’s modify the system of equations stated in Question 28-6 in a different way. Suppose we
subtract 12 from the right side of the second equation, producing this two-by-two system:

y= (x+ 1)^2

x

y

(0,1)

Figure 30-7 Illustration for Answer 28-6. The first function

is graphed as a solid curve; the second function

is graphed as a dashed curve. The real-number

solution appears as a point where the curves

intersect. On the x axis, each increment is 1/2 unit.

On the y axis, each increment is 2 units.