SOLUTION: Find the vertex and the axis of symmetry for the following graph: {{{f(x)=-x^2+2x+3}}}

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: Find the vertex and the axis of symmetry for the following graph: {{{f(x)=-x^2+2x+3}}}      Log On


   

Question 131105: Find the vertex and the axis of symmetry for the following graph:


f%28x%29=-x%5E2%2B2x%2B3
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
To find the vertex, we first need to find the line of symmetry (ie the x-coordinate of the vertex)
To find the line of symmetry, use this formula:

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

From the equation y=-x%5E2%2B2x%2B3 we can see that a=-1 and b=2

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


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



x=1 Reduce


So the line of symmetry is x=1


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


Lets evaluate f%281%29

f%28x%29=-x%5E2%2B2x%2B3 Start with the given polynomial


f%281%29=-%281%29%5E2%2B2%281%29%2B3 Plug in x=1


f%281%29=-%281%29%2B2%281%29%2B3 Raise 1 to the second power to get 1


f%281%29=-%281%29%2B2%2B3 Multiply 2 by 1 to get 2


f%281%29=4 Now combine like terms


So the vertex is (1,4)



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Answer:

So the line of symmetry is x=1 and the vertex is (1,4)



If we graph, we can visually verify our answer

Graph of f%28x%29=-x%5E2%2B2x%2B3 with the line of symmetry x=1 and the vertex (1,4)