Find the roots and use them to graph the related function.
{ x| - 3 ≤ x ≤ 2 }
Practice makes perfect
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.
x=2± 10/- 4
x=2 + 10/- 4
x=2 - 10/- 4
x=- 2/4-10/4
x=- 2/4+10/4
x=- 3
x= 2
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.
