Rewrite fx = 3x² + 22x + 1 from vertex form to standard form

To convert the given quadratic function from vertex form to standard form, we first need to understand what both forms look like.

The vertex form of a quadratic function is often written as:

f(x) = a(x – h)² + k

Where (h, k) is the vertex of the parabola. The standard form is typically presented as:

f(x) = ax² + bx + c

The given function is already in standard form: f(x) = 3x² + 22x + 1. However, if we want to express it in vertex form, we first factor out the coefficient of x² and complete the square:

1. Start with the function: f(x) = 3x² + 22x + 1
2. Factor out the coefficient of x² from the first two terms:
3. f(x) = 3(x² + rac{22}{3}x) + 1
4. To complete the square inside the parentheses, take half of the coefficient of x, square it, and add it inside the parentheses:
5. Half of rac{22}{3} is rac{11}{3}, and squaring it gives rac{121}{9}.
6. Now we adjust the equation:
7. f(x) = 3igg(x² + rac{22}{3}x + rac{121}{9} – rac{121}{9}igg) + 1
8. Rewrite it as:
9. f(x) = 3igg((x + rac{11}{3})² – rac{121}{9}igg) + 1
10. Now distribute the 3:
11. f(x) = 3(x + rac{11}{3})² – rac{363}{9} + 1
12. Convert 1 into a fraction with a denominator of 9:
13. f(x) = 3(x + rac{11}{3})² – rac{363}{9} + rac{9}{9}
14. Combine the constants:
15. f(x) = 3(x + rac{11}{3})² – rac{354}{9}

So, the function can therefore be rewritten in vertex form as:

f(x) = 3(x + rac{11}{3})² – rac{354}{9}

In summary, we have effectively expressed the quadratic function in another form, but since our original request was to rewrite it in standard form, we confirm that it originally was already in standard form: f(x) = 3x² + 22x + 1.