To rewrite the quadratic function from vertex form to standard form, we first need to understand what vertex form looks like. Vertex form of a quadratic function is generally expressed as:
f(x) = a(x – h)² + k
where (h, k) is the vertex of the parabola.
In this case, your function is f(x) = 2x² + 12x + 3. To convert it to standard form, we’ll complete the square.
- Factor out the coefficient of x² from the first two terms: f(x) = 2(x² + 6x) + 3
- Next, to complete the square inside the parentheses, take half of the coefficient of x (which is 6), square it, and add and subtract this value inside the brackets:
Half of 6 is 3, and 3² is 9. So we adjust the equation:
f(x) = 2(x² + 6x + 9 – 9) + 3
This simplifies to:
f(x) = 2((x + 3)² – 9) + 3
- Distribute the 2: f(x) = 2(x + 3)² – 18 + 3
- This results in f(x) = 2(x + 3)² – 15
Now, we can express the function in the standard form by leaving it as it is:
Final answer in standard form is: f(x) = 2(x + 3)² – 15
Note: Standard form is often also written as f(x) = ax² + bx + c. In this case, the function can be represented after expanding the square, but the completed square gives us the vertex position clearly.