SOLUTION: f(x) = -3x^2 -10x -2 What is the vertex? Where is the symmetric point?

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: f(x) = -3x^2 -10x -2 What is the vertex? Where is the symmetric point?      Log On


   

Question 750270: f(x) = -3x^2 -10x -2
What is the vertex?
Where is the symmetric point?
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

vertex form:
f%28x%29+=+a%28x+-h%29%5E2+%2B+k
vertex: (h,k)
axis of symmetry: x+=+h

given:
f%28x%29+=+-3x%5E2+-10x+-2...write it in vertex form
f%28x%29+=+%28-3x%5E2+-10x%2B_%29+-2..complete the square...recall that f%28x%29=y

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


y=-3+x%5E2-10+x-2 Start with the given equation



y%2B2=-3+x%5E2-10+x Add 2 to both sides



y%2B2=-3%28x%5E2%2B%2810%2F3%29x%29 Factor out the leading coefficient -3



Take half of the x coefficient 10%2F3 to get 5%2F3 (ie %281%2F2%29%2810%2F3%29=5%2F3).


Now square 5%2F3 to get 25%2F9 (ie %285%2F3%29%5E2=%285%2F3%29%285%2F3%29=25%2F9)





y%2B2=-3%28x%5E2%2B%2810%2F3%29x%2B25%2F9-25%2F9%29 Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of 25%2F9 does not change the equation




y%2B2=-3%28%28x%2B5%2F3%29%5E2-25%2F9%29 Now factor x%5E2%2B%2810%2F3%29x%2B25%2F9 to get %28x%2B5%2F3%29%5E2



y%2B2=-3%28x%2B5%2F3%29%5E2%2B3%2825%2F9%29 Distribute



y%2B2=-3%28x%2B5%2F3%29%5E2%2B25%2F3 Multiply



y=-3%28x%2B5%2F3%29%5E2%2B25%2F3-2 Now add %2B2 to both sides to isolate y



y=-3%28x%2B5%2F3%29%5E2%2B19%2F3 Combine like terms




Now the quadratic is in vertex form y=a%28x-h%29%5E2%2Bk where a=-3, h=-5%2F3, and k=19%2F3. Remember (h,k) is the vertex and "a" is the stretch/compression factor.




Check:


Notice if we graph the original equation y=-3x%5E2-10x-2 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C-3x%5E2-10x-2%29 Graph of y=-3x%5E2-10x-2. Notice how the vertex is (-5%2F3,19%2F3).



Notice if we graph the final equation y=-3%28x%2B5%2F3%29%5E2%2B19%2F3 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C-3%28x%2B5%2F3%29%5E2%2B19%2F3%29 Graph of y=-3%28x%2B5%2F3%29%5E2%2B19%2F3. Notice how the vertex is also (-5%2F3,19%2F3).



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






since h=-5%2F3, axis of symmetry: x+=+-5%2F3

here is a graph with your parabola and its axis of symmetry: