SOLUTION: Find the vertex, and the x and y intercepts by completing the following: f(x) = -x^2 + 6x - 8

Algebra ->  Algebra  -> Functions -> SOLUTION: Find the vertex, and the x and y intercepts by completing the following: f(x) = -x^2 + 6x - 8       Log On

Ad: Algebrator™ solves your algebra problems and provides step-by-step explanations!
Ad: Algebra Solved!™: algebra software solves algebra homework problems with step-by-step help!

   

Question 104962: Find the vertex, and the x and y intercepts by completing the following:
f(x) = -x^2 + 6x - 8

Answer by MathLover1(6611) About Me  (Show Source):
You can put this solution on YOUR website!

you can use y+instead of f%28x%29+
y+=+-x%5E2+%2B+6x+-+8

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


y=-1+x%5E2%2B6+x-8 Start with the given equation



y%2B8=-1+x%5E2%2B6+x Add 8 to both sides



y%2B8=-1%28x%5E2-6x%29 Factor out the leading coefficient -1



Take half of the x coefficient -6 to get -3 (ie %281%2F2%29%28-6%29=-3).


Now square -3 to get 9 (ie %28-3%29%5E2=%28-3%29%28-3%29=9)





y%2B8=-1%28x%5E2-6x%2B9-9%29 Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of 9 does not change the equation




y%2B8=-1%28%28x-3%29%5E2-9%29 Now factor x%5E2-6x%2B9 to get %28x-3%29%5E2



y%2B8=-1%28x-3%29%5E2%2B1%289%29 Distribute



y%2B8=-1%28x-3%29%5E2%2B9 Multiply



y=-1%28x-3%29%5E2%2B9-8 Now add %2B8 to both sides to isolate y



y=-1%28x-3%29%5E2%2B1 Combine like terms




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




Check:


Notice if we graph the original equation y=-1x%5E2%2B6x-8 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C-1x%5E2%2B6x-8%29 Graph of y=-1x%5E2%2B6x-8. Notice how the vertex is (3,1).



Notice if we graph the final equation y=-1%28x-3%29%5E2%2B1 we get:


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



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