SOLUTION: Find the axis of symmetry... 1. y=-x^2+4x+2 2. y=x^2+x+1 Find the x-intercepts... 1.y=x^2+2x-8 2. y=x^2-5x-10

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: Find the axis of symmetry... 1. y=-x^2+4x+2 2. y=x^2+x+1 Find the x-intercepts... 1.y=x^2+2x-8 2. y=x^2-5x-10      Log On


   

Question 86122: Find the axis of symmetry...
1. y=-x^2+4x+2
2. y=x^2+x+1
Find the x-intercepts...
1.y=x^2+2x-8
2. y=x^2-5x-10
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
1.
Solved by pluggable solver: Completing the Square to Get a Quadratic into Vertex Form


y=-1+x%5E2%2B4+x%2B2 Start with the given equation



y-2=-1+x%5E2%2B4+x Subtract 2 from both sides



y-2=-1%28x%5E2-4x%29 Factor out the leading coefficient -1



Take half of the x coefficient -4 to get -2 (ie %281%2F2%29%28-4%29=-2).


Now square -2 to get 4 (ie %28-2%29%5E2=%28-2%29%28-2%29=4)





y-2=-1%28x%5E2-4x%2B4-4%29 Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of 4 does not change the equation




y-2=-1%28%28x-2%29%5E2-4%29 Now factor x%5E2-4x%2B4 to get %28x-2%29%5E2



y-2=-1%28x-2%29%5E2%2B1%284%29 Distribute



y-2=-1%28x-2%29%5E2%2B4 Multiply



y=-1%28x-2%29%5E2%2B4%2B2 Now add 2 to both sides to isolate y



y=-1%28x-2%29%5E2%2B6 Combine like terms




Now the quadratic is in vertex form y=a%28x-h%29%5E2%2Bk where a=-1, h=2, and k=6. Remember (h,k) is the vertex and "a" is the stretch/compression factor.




Check:


Notice if we graph the original equation y=-1x%5E2%2B4x%2B2 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C-1x%5E2%2B4x%2B2%29 Graph of y=-1x%5E2%2B4x%2B2. Notice how the vertex is (2,6).



Notice if we graph the final equation y=-1%28x-2%29%5E2%2B6 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C-1%28x-2%29%5E2%2B6%29 Graph of y=-1%28x-2%29%5E2%2B6. Notice how the vertex is also (2,6).



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






2.
Solved by pluggable solver: Completing the Square to Get a Quadratic into Vertex Form


y=1+x%5E2%2B1+x%2B1 Start with the given equation



y-1=1+x%5E2%2B1+x Subtract 1 from both sides



y-1=1%28x%5E2%2B1x%29 Factor out the leading coefficient 1



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


Now square 1%2F2 to get 1%2F4 (ie %281%2F2%29%5E2=%281%2F2%29%281%2F2%29=1%2F4)





y-1=1%28x%5E2%2B1x%2B1%2F4-1%2F4%29 Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of 1%2F4 does not change the equation




y-1=1%28%28x%2B1%2F2%29%5E2-1%2F4%29 Now factor x%5E2%2B1x%2B1%2F4 to get %28x%2B1%2F2%29%5E2



y-1=1%28x%2B1%2F2%29%5E2-1%281%2F4%29 Distribute



y-1=1%28x%2B1%2F2%29%5E2-1%2F4 Multiply



y=1%28x%2B1%2F2%29%5E2-1%2F4%2B1 Now add 1 to both sides to isolate y



y=1%28x%2B1%2F2%29%5E2%2B3%2F4 Combine like terms




Now the quadratic is in vertex form y=a%28x-h%29%5E2%2Bk where a=1, h=-1%2F2, and k=3%2F4. Remember (h,k) is the vertex and "a" is the stretch/compression factor.




Check:


Notice if we graph the original equation y=1x%5E2%2B1x%2B1 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C1x%5E2%2B1x%2B1%29 Graph of y=1x%5E2%2B1x%2B1. Notice how the vertex is (-1%2F2,3%2F4).



Notice if we graph the final equation y=1%28x%2B1%2F2%29%5E2%2B3%2F4 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C1%28x%2B1%2F2%29%5E2%2B3%2F4%29 Graph of y=1%28x%2B1%2F2%29%5E2%2B3%2F4. Notice how the vertex is also (-1%2F2,3%2F4).



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






We can find the x-intercepts by the quadratic formula
1.
Starting with the general quadratic

ax%5E2%2Bbx%2Bc

the general form of the quadratic equation is:

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

So lets solve x%5E2%2B2%2Ax-8

x+=+%28-2+%2B-+sqrt%28+%282%29%5E2-4%2A1%2A-8+%29%29%2F%282%2A1%29 Plug in a=1, b=2, and c=-8



x+=+%28-2+%2B-+sqrt%28+4-4%2A1%2A-8+%29%29%2F%282%2A1%29 Square 2 to get 4



x+=+%28-2+%2B-+sqrt%28+4%2B32+%29%29%2F%282%2A1%29 Multiply -4%2A-8%2A1 to get 32



x+=+%28-2+%2B-+sqrt%28+36+%29%29%2F%282%2A1%29 Combine like terms in the radicand (everything under the square root)



x+=+%28-2+%2B-+6%29%2F%282%2A1%29 Simplify the square root



x+=+%28-2+%2B-+6%29%2F2 Multiply 2 and 1 to get 2

So now the expression breaks down into two parts

x+=+%28-2+%2B+6%29%2F2 or x+=+%28-2+-+6%29%2F2

Lets look at the first part:

x=4%2F2 Add the terms in the numerator
x=2 Divide

So one answer is
x=2
Now lets look at the second part:

x=-8%2F2 Subtract the terms in the numerator
x=-4 Divide

So another answer is
x=-4

So our solutions are:
x=2 or x=-4

Notice when we graph x%5E2%2B2%2Ax-8 we get:

+graph%28+500%2C+500%2C+-14%2C+12%2C+-14%2C+12%2C1%2Ax%5E2%2B2%2Ax%2B-8%29+

and we can see that the roots are x=2 and x=-4. This verifies our answer

2.
Starting with the general quadratic

ax%5E2%2Bbx%2Bc

the general form of the quadratic equation is:

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

So lets solve x%5E2-5%2Ax-10

x+=+%285+%2B-+sqrt%28+%28-5%29%5E2-4%2A1%2A-10+%29%29%2F%282%2A1%29 Plug in a=1, b=-5, and c=-10



x+=+%285+%2B-+sqrt%28+25-4%2A1%2A-10+%29%29%2F%282%2A1%29 Square -5 to get 25



x+=+%285+%2B-+sqrt%28+25%2B40+%29%29%2F%282%2A1%29 Multiply -4%2A-10%2A1 to get 40



x+=+%285+%2B-+sqrt%28+65+%29%29%2F%282%2A1%29 Combine like terms in the radicand (everything under the square root)



x+=+%285+%2B-+sqrt%2865%29%29%2F%282%2A1%29 Simplify the square root



x+=+%285+%2B-+sqrt%2865%29%29%2F2 Multiply 2 and 1 to get 2

So now the expression breaks down into two parts

x+=+%285+%2B+sqrt%2865%29%29%2F2 or x+=+%285+-+sqrt%2865%29%29%2F2


Which approximate to

x=6.53112887414927 or x=-1.53112887414927


So our solutions are:
x=6.53112887414927 or x=-1.53112887414927

Notice when we graph x%5E2-5%2Ax-10 we get:



when we use the root finder feature on our calculator, we find that x=6.53112887414927 and x=-1.53112887414927.So this verifies our answer