Exercise 28 Page 211

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= 3x^2 + 12x + 36 We see that a= 3, b= 12, and c= 36. Let's substitute these values into the Quadratic Formula to find the roots of 3x^2+12x+36=0.

We have found a negative discriminant, - 16. Therefore, the related quadratic function has no real solutions. This means the graph of the quadratic function does not intercept the x-axis. 3x^2+12x+36≤ 0 The solution of the given quadratic inequality consists of x-values for which the graph of the related quadratic function lies on and below the x-axis. The graph opens upward since a= 3 is greater than zero.

We see that the graph lies above the x-axis for all x-values. Hence, there is no x-value satisfying 3x^2+12x+36≤ 0. The solution set of the given quadratic inequality is an empty set. { } ⇔ ∅

Showing Our Work

Drawing the parabola using a point on the curve and vertex

We need at least three points to draw the graph of the related quadratic function.To draw the graph, we will first find the vertex, (- b2a,f(- b2a)).Then, we will find a point on the curve and reflect it across the axis of symmetry. Let's start by calculating the x-coordinate of the vertex. To do so, we will substitute a= 3 and b= 12 into - b2a.

- b/2a

SubstituteII

a= 3, b= 12

- 12/2( 3)

Multiply

Multiply

- 12/6

CalcQuot

Calculate quotient

- 2

To find the y-coordinate of the vertex, we will substitute - 2 for x in the related function y=3x^2+12x+36.

y=3x^2+12x+36

Substitute

x= - 2

y=3(- 2)^2+12( - 2) +36

▼

Simplify right-hand side

CalcPow

Calculate power

y=3(4)+12(- 2) +36

Multiply

Multiply

y=12-24 +36

AddSubTerms

Add and subtract terms

y=24

The vertex is (- 2,24). Note that the formula for the x-coordinate of the vertex is the same as the formula for the axis of symmetry, which is x=- 2. Now, we will choose an x-value and substitute it for x in the function. Let's try x=1.

y=3x^2+12x+36

Substitute

x= 1

y=3( 1)^2+12( 1)+36

▼

Simplify right-hand side

BaseOne

1^a=1

y=3(1)+12(1)+36

Multiply

Multiply

y=3+12+36

AddTerms

Add terms

y=51

The point (1,51) lies on the curve. Since the axis of symmetry divides the graph into two mirror images, there exists another point on the oppositve side of the axis of symmetry with the same y-value.

Point	Reflection Across x=- 2
(1, 51)	(- 5, 51)

The vertex, along with the points, is helpful to graph a parabola.