SOLUTION: Sketch the graph of f(x) = x^2-6x +8. Label all x-intercept, y-intercepts and the vertex.

Algebra ->  Graphs -> SOLUTION: Sketch the graph of f(x) = x^2-6x +8. Label all x-intercept, y-intercepts and the vertex.      Log On


   

Question 128198: Sketch the graph of f(x) = x^2-6x +8. Label all x-intercept, y-intercepts
and the vertex.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
f%28x%29+=+x%5E2-6x+%2B8 Start with the given function


To find the x-intercept, let f%28x%29=0

0=+x%5E2-6x+%2B8 Plug in f%28x%29=0



0=%28x-4%29%28x-2%29 Factor the right side (note: if you need help with factoring, check out this solver)



Now set each factor equal to zero:
x-4=0 or x-2=0

x=4 or x=2 Now solve for x in each case



So the x-intercepts are (2,0) and (4,0)



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To find the y-intercept, plug in x=0


f%28x%29+=+x%5E2-6x+%2B8 Start with the given function


f%280%29=1%280%29%5E2-6%280%29%2B8 Plug in x=0


f%280%29=1%2A0-6%2A0%2B8 Raise 0 to the 2nd power to get 0


f%280%29=0-6%2A0%2B8 Multiply 1 and 0 to get 0


f%280%29=0-0%2B8 Multiply 6 and 0 to get 0


f%280%29=0%2B8 Subtract 0 from 0 to get 0


f%280%29=8 Add 0 and 8 to get 8


So when x=0, we have y=8



So this means that the y-intercept is (0,8)





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To find the vertex, we first need to find 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=x%5E2-6x%2B8 we can see that a=1 and b=-6

x=%28--6%29%2F%282%2A1%29 Plug in b=-6 and a=1


x=6%2F%282%2A1%29 Negate -6 to get 6


x=%286%29%2F2 Multiply 2 and 1 to get 2



x=3 Reduce


So the axis of symmetry is x=3


So the x-coordinate of the vertex is x=3. Lets plug this into the equation to find the y-coordinate of the vertex.


Lets evaluate f%283%29

f%28x%29=x%5E2-6x%2B8 Start with the given polynomial


f%283%29=%283%29%5E2-6%283%29%2B8 Plug in x=3


f%283%29=%289%29-6%283%29%2B8 Raise 3 to the second power to get 9


f%283%29=%289%29-18%2B8 Multiply 6 by 3 to get 18


f%283%29=-1 Now combine like terms


So the vertex is (3,-1)




So putting all of this together, we get


+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C+x%5E2-6x%2B8%29+ Graph of f%28x%29=x%5E2-6x%2B8


and we can see that the x-intercepts are (2,0) and (4,0), the graph has the y-intercept (0,8), and the vertex is (3,-1). So this visually verifies our answer.