SOLUTION: f (x)=6x^2+x-2. Find the y and X intercepts Determine the maximum or the minimum point of the function Sketch the graph of f (x)=6x^2+x-2

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equation Customizable Word Problems -> SOLUTION: f (x)=6x^2+x-2. Find the y and X intercepts Determine the maximum or the minimum point of the function Sketch the graph of f (x)=6x^2+x-2      Log On


   

Question 987386: f (x)=6x^2+x-2.
Find the y and X intercepts
Determine the maximum or the minimum point of the function
Sketch the graph of f (x)=6x^2+x-2
Answer by macston(5194) About Me  (Show Source):
You can put this solution on YOUR website!
.
f%28x%29=6x%5E2%2Bx-2
.
Set x=0 to find y-intercept:
f%280%29=6%280%29%5E2%2B0-2
y=-2 ANSWER: The y intercept is (0,-2)
.
Set y=0 to find x-intercept:
0=6x^2+x-2
.
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 6x%5E2%2B1x%2B-2+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%281%29%5E2-4%2A6%2A-2=49.

Discriminant d=49 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-1%2B-sqrt%28+49+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%281%29%2Bsqrt%28+49+%29%29%2F2%5C6+=+0.5
x%5B2%5D+=+%28-%281%29-sqrt%28+49+%29%29%2F2%5C6+=+-0.666666666666667

Quadratic expression 6x%5E2%2B1x%2B-2 can be factored:
6x%5E2%2B1x%2B-2+=+6%28x-0.5%29%2A%28x--0.666666666666667%29
Again, the answer is: 0.5, -0.666666666666667. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+6%2Ax%5E2%2B1%2Ax%2B-2+%29

.
ANSWER: The x intercepts are (0.5,0) and (-2/3,0)
.
Derivatives:
.
f'%28x%29=12x%2B1 Set this to zero to find critical point.
f''%28x%29=12 This is positive so we will find the minimum.
.
Find critical point:
Set first derivative=0 to find x value of critical point
f'%28x%29=12x%2B1
0=12x%2B1
-1=12x
-1%2F12=x
.
Solve for f(-1/12) to find the y value of critical point:
f%28x%29=6x%5E2%2Bx-2
y=6%28-1%2F12%29%5E2-1%2F12-2
y=6%2F144-12%2F144-2
y=-2 6/144=-2 1/24
.
Critical point (turning point, minumum): (-1/12,-2 1/24)
.
GRAPH:
.
+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C+6x%5E2%2Bx-2%29+