Question 683304: what is the vertex form of y=3x^2+12x+20 ?
Answer by RedemptiveMath(80)   (Show Source):
You can put this solution on YOUR website! The vertex is the highest or lowest point of the parabola depending on a certain number. There are multiple approaches you can take for this quadratic function. We can see that y=3x^2+12x+20 has a=3 and b=12 in the common structure ax^2+bx+c. Using the shortcut approach, we have
x = -(b/2a) = -(12/6) = -2.
Plugging -2 in for x in the original equation:
3(-2)^2+12(-2)+20 = 3(4)+(-24)+20 = 12-24+20= 8.
The x-coordinate of the vertex is -2 and the y-coordinate is 8. Since a is positive (the parabola would open upwards), the minimum (lowest) value is the point (-2, 8), which this is also the vertex.
The technical approach, of which this shortcut approach is derived from, is formatted as y = a(x – h)^2 + k, where (h, k) is the vertex and "a" is the same as the "a" in the structure ax^2+bx+c. However, most textbook problems write the quadratic in the latter form. To get from the standard form to the vertex form, the process known as "completing the square" can be used. This process is described below using your problem.
y = 3x^2+12x+20 (standard form)
y-20 = 3x^2+12x (move the constant c to the other side)
y-20 = 3(x^2+4x) (factor out the largest coefficient number on the right side)
y-20+3 = 3(x^2+4x) (if a does not equal 1, copy it to the end of the left side)
y-20+3(4) = 3(x^2+4x+4) (half the coefficient of x term, square it, multiply the new number to the last number of the left side, and add it to the inside of the parentheses of the right side)
y-8 = 3(x+2)^2 (combine like terms on the left side and rewrite the quantity on the right side as a binomial square)
y = 3[x-(-2)]^2 + 8 (move the constant from the left side to the right, and change the operation inside the parentheses so that it makes subtraction)
Note that you must compare the two forms listed above this work to see where a, b, and c are and how they change accordingly. Although more complex, we can see from this form that h = -2 and k = 8. You must be wary of all signs when using this method. You cannot change any sign from the standard form randomly. You must use algebra to rewrite the equation correctly.
Using both methods, we can see that the vertex is (-2, 8).
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