SOLUTION: Find the vertex and the axis of symmetry for the following graph: {{{f(x)=3x^2-24x+51}}}

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)=3x^2-24x+51}}}      Log On


   

Question 131104: Find the vertex and the axis of symmetry for the following graph:


f%28x%29=3x%5E2-24x%2B51
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=3x%5E2-24x%2B51 we can see that a=3 and b=-24

x=%28--24%29%2F%282%2A3%29 Plug in b=-24 and a=3


x=24%2F%282%2A3%29 Negate -24 to get 24


x=%2824%29%2F6 Multiply 2 and 3 to get 6



x=4 Reduce


So the line of symmetry is x=4


So the x-coordinate of the vertex is x=4. Lets plug this into the equation to find the y-coordinate of the vertex.


Lets evaluate f%284%29

f%28x%29=3x%5E2-24x%2B51 Start with the given polynomial


f%284%29=3%284%29%5E2-24%284%29%2B51 Plug in x=4


f%284%29=3%2816%29-24%284%29%2B51 Raise 4 to the second power to get 16


f%284%29=48-24%284%29%2B51 Multiply 3 by 16 to get 48


f%284%29=48-96%2B51 Multiply 24 by 4 to get 96


f%284%29=3 Now combine like terms


So the vertex is (4,3)




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Answer:


So the line of symmetry is x=4 and the vertex is (4,3)



If we graph, we can visually verify our answer

Graph of f%28x%29=3x%5E2-24x%2B51 with the line of symmetry x=4 and the vertex (4,3)