SOLUTION: Someone PLEASE help me! I am solving and graphing parabolas and I just dont understand the steps involved....I also need to graph each one and find the vertex's and intercepts?? I

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: Someone PLEASE help me! I am solving and graphing parabolas and I just dont understand the steps involved....I also need to graph each one and find the vertex's and intercepts?? I       Log On


   

Question 111879: Someone PLEASE help me! I am solving and graphing parabolas and I just dont understand the steps involved....I also need to graph each one and find the vertex's and intercepts?? I dont know what to do first...no has explained this to me so I can understand. Here is an example of what I am working on: f(x)=-3x^2 + x -5
I would appreciate any help!
Thanks so much!
Bonnie Cook
Found 2 solutions by jim_thompson5910, MathLover1:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
First let's find the vertex. To find the vertex, we first need the axis of symmetry (ie the x-coordinate of the vertex)


To find the axis of symmetry, use this formula:

x=-b%2F%282a%29

From the equation y=-3x%5E2%2Bx-5 we can see that a=-3 and b=1

x=%28-1%29%2F%282%2A-3%29 Plug in b=1 and a=-3


x=%28-1%29%2F-6 Multiply 2 and -3 to get -6



x=1%2F6 Reduce


So the axis of symmetry is x=1%2F6


So the x-coordinate of the vertex is x=1%2F6. Lets plug this into the equation to find the y-coordinate of the vertex.

Lets evaluate f%281%2F6%29

f%28x%29=-3x%5E2%2Bx-5 Start with the given polynomial


f%281%2F6%29=-3%281%2F6%29%5E2%2B%281%2F6%29-5 Plug in x=1%2F6


f%281%2F6%29=-3%281%2F36%29%2B%281%2F6%29-5 Square 1%2F6 to get 1%2F36


f%281%2F6%29=-3%2F36%2B%281%2F6%29-5 Multiply



f%281%2F6%29=-1%2F12%2B1%2F6-5 Reduce


f%281%2F6%29=-59%2F12 Combine like terms


So the y-coordinate of the vertex is y=-59%2F12

So the vertex is



Now let's find the x-intercepts. To find the x-intercepts, let y=0 and solve for x

0=-3x%5E2%2Bx-5 Set y equal to zero to solve for x


Solved by pluggable solver: Quadratic Formula
Let's use the quadratic formula to solve for x:


Starting with the general quadratic


ax%5E2%2Bbx%2Bc=0


the general solution using the quadratic equation is:


x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29




So lets solve -3%2Ax%5E2%2Bx-5=0 ( notice a=-3, b=1, and c=-5)





x+=+%28-1+%2B-+sqrt%28+%281%29%5E2-4%2A-3%2A-5+%29%29%2F%282%2A-3%29 Plug in a=-3, b=1, and c=-5




x+=+%28-1+%2B-+sqrt%28+1-4%2A-3%2A-5+%29%29%2F%282%2A-3%29 Square 1 to get 1




x+=+%28-1+%2B-+sqrt%28+1%2B-60+%29%29%2F%282%2A-3%29 Multiply -4%2A-5%2A-3 to get -60




x+=+%28-1+%2B-+sqrt%28+-59+%29%29%2F%282%2A-3%29 Combine like terms in the radicand (everything under the square root)




x+=+%28-1+%2B-+i%2Asqrt%2859%29%29%2F%282%2A-3%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)




x+=+%28-1+%2B-+i%2Asqrt%2859%29%29%2F%28-6%29 Multiply 2 and -3 to get -6




After simplifying, the quadratic has roots of


x=1%2F6-sqrt%2859%29%2F6%2Ai or x=1%2F6%2Bsqrt%2859%29%2F6%2Ai

Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

Solved by pluggable solver: Completing the Square to Get a Quadratic into Vertex Form


y=-3+x%5E2%2B1+x-5 Start with the given equation



y%2B5=-3+x%5E2%2B1+x Add 5 to both sides



y%2B5=-3%28x%5E2%2B%28-1%2F3%29x%29 Factor out the leading coefficient -3



Take half of the x coefficient -1%2F3 to get -1%2F6 (ie %281%2F2%29%28-1%2F3%29=-1%2F6).


Now square -1%2F6 to get 1%2F36 (ie %28-1%2F6%29%5E2=%28-1%2F6%29%28-1%2F6%29=1%2F36)





y%2B5=-3%28x%5E2%2B%28-1%2F3%29x%2B1%2F36-1%2F36%29 Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of 1%2F36 does not change the equation




y%2B5=-3%28%28x-1%2F6%29%5E2-1%2F36%29 Now factor x%5E2%2B%28-1%2F3%29x%2B1%2F36 to get %28x-1%2F6%29%5E2



y%2B5=-3%28x-1%2F6%29%5E2%2B3%281%2F36%29 Distribute



y%2B5=-3%28x-1%2F6%29%5E2%2B1%2F12 Multiply



y=-3%28x-1%2F6%29%5E2%2B1%2F12-5 Now add %2B5 to both sides to isolate y



y=-3%28x-1%2F6%29%5E2-59%2F12 Combine like terms




Now the quadratic is in vertex form y=a%28x-h%29%5E2%2Bk where a=-3, h=1%2F6, and k=-59%2F12. Remember (h,k) is the vertex and "a" is the stretch/compression factor.




Check:


Notice if we graph the original equation y=-3x%5E2%2B1x-5 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C-3x%5E2%2B1x-5%29 Graph of y=-3x%5E2%2B1x-5. Notice how the vertex is (1%2F6,-59%2F12).



Notice if we graph the final equation y=-3%28x-1%2F6%29%5E2-59%2F12 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C-3%28x-1%2F6%29%5E2-59%2F12%29 Graph of y=-3%28x-1%2F6%29%5E2-59%2F12. Notice how the vertex is also (1%2F6,-59%2F12).



So if these two equations were graphed on the same coordinate plane, one would overlap another perfectly. So this visually verifies our answer.






as you can see on graph above, there are no x_intercepts
now, use the quadratic formula to solve for x to show that this function has no+real+roots

Solved by pluggable solver: Quadratic Formula
Let's use the quadratic formula to solve for x:


Starting with the general quadratic


ax%5E2%2Bbx%2Bc=0


the general solution using the quadratic equation is:


x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29




So lets solve -3%2Ax%5E2%2Bx-5=0 ( notice a=-3, b=1, and c=-5)





x+=+%28-1+%2B-+sqrt%28+%281%29%5E2-4%2A-3%2A-5+%29%29%2F%282%2A-3%29 Plug in a=-3, b=1, and c=-5




x+=+%28-1+%2B-+sqrt%28+1-4%2A-3%2A-5+%29%29%2F%282%2A-3%29 Square 1 to get 1




x+=+%28-1+%2B-+sqrt%28+1%2B-60+%29%29%2F%282%2A-3%29 Multiply -4%2A-5%2A-3 to get -60




x+=+%28-1+%2B-+sqrt%28+-59+%29%29%2F%282%2A-3%29 Combine like terms in the radicand (everything under the square root)




x+=+%28-1+%2B-+i%2Asqrt%2859%29%29%2F%282%2A-3%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)




x+=+%28-1+%2B-+i%2Asqrt%2859%29%29%2F%28-6%29 Multiply 2 and -3 to get -6




After simplifying, the quadratic has roots of


x=1%2F6-sqrt%2859%29%2F6%2Ai or x=1%2F6%2Bsqrt%2859%29%2F6%2Ai