Exercise 132 Page 251

To graph the inequalities, we will begin by considering their related equations. This will allow us to graph their boundary curve. Inequalities:& y ≥ x^2 y ≥ (x-4)^2+2 [1em] Equations:& y = x^2 y = (x-4)^2+2

Graphing the Boundaries

The first equation is the parent function of the quadratic function family. The second equation is a translation of the parent function by 4 units right and 2 units up. Since the inequalities are both nonstrict, the boundary curve is a part of the solutions set and should be solid.

Shading

To determine which side of the boundaries we should shade, we must test a point in the inequality that is not on any of the boundaries. We can, for example, test (0,1).

y≥ x^2

SubstituteII

x= 0, y= 1

1? ≥ 0^2

CalcPow

Calculate power

1≥ 0 ✓

For the first inequality, we should shade the side that contains the test point.

Next, we will test the second inequality.

y≥(x-4)^2+2

SubstituteII

x= 0, y= 1

1? ≥ ( 0-4)^2+2

▼

Evaluate right-hand side

SubTerm

Subtract term

1? ≥ (-4)^2+2

NegBaseToPosBase

(- a)^2=a^2

1? ≥ 16+2

AddTerms

Add terms

1 ≱ 18 *

For our second inequality, we should shade the side of the curve that does not contain the test point.

Finally, we will isolate the overlapping area to obtain the solution set.