Exercise 6 Page 210

We will start by sketching the related quadratic function. To do so, we first need to identify the values of a, b, and c. y=- 2x^2-2x+12 ⇔ y= - 2x^2+( - 2)x+ 12 We see that a= - 2, b= - 2, and c= 12. Let's substitute these values into the Quadratic Formula to find the roots of - 2x^2-2x+12=0.

Now we can calculate the first root using the positive sign and the second root using the negative sign.

The solution of the given quadratic inequality, -2x^2-2x+12≥0, consists of x-values for which the graph of the related quadratic function lies on and above the x-axis. The graph opens downwards, since a= - 2 is less than zero.

We see that the graph lies on and above the x-axis between x=- 3 and x=2. { x| - 3 ≤ x ≤ 2 } ⇕ [- 3, 2]

Showing Our Work

Drawing the parabola using roots and vertex

We already know the roots, which are around - 3 and 2. To draw the graph, we will find the vertex, (- b2a,f(- b2a)). Let's start by calculating its x-coordinate. To do so, we will substitute a= - 2 and b= - 2 into - b2a.

- b/2a

SubstituteII

a= - 2, b= - 2

- - 2/2( - 2)

MultPosNeg

a(- b)=- a * b

- - 2/- 4

DivNegNeg

- a/- b=a/b

- 2/4

CalcQuot

Calculate quotient

- 0.5

Finally, to find the y-coordinate of the vertex, we will substitute - 0.5 for x in the related function y=- 2x^2-2x+12.

y=- 2x^2-2x+12

Substitute

x= - 0.5

y=- 2(- 0.5)^2-2( - 0.5) +12

▼

Simplify right-hand side

CalcPow

Calculate power

y=- 2(0.25)-2(- 0.5) +12

Multiply

Multiply

y=- 0.5 +1 +12

AddSubTerms

Add and subtract terms

y=12.5

The vertex is (- 0.5,12.5 ). This point, along with the roots, is helpful to graph a parabola.