Question

Transform the quadratic equation into standard form and find the y-intercept.

y
=
3
(
x
+
2
)
2
+
3

Answer 1

To transform the given quadratic equation into standard form and find the y-intercept, we'll follow these steps:

1. **Given equation**:
\[
y = 3(x + 2)^2 + 3
\]

2. **Expand the equation**: First, expand \((x + 2)^2\).

\[
(x + 2)^2 = x^2 + 4x + 4
\]

3. **Substitute back** into the equation:

\[
y = 3(x^2 + 4x + 4) + 3
\]

4. **Distribute the 3**:

\[
y = 3x^2 + 12x + 12 + 3
\]

5. **Combine like terms**:

\[
y = 3x^2 + 12x + 15
\]

Now, the equation has been transformed into standard form:

\[
y = 3x^2 + 12x + 15
\]

### Finding the y-intercept:

To find the y-intercept, set \(x = 0\) and solve for \(y\):

\[
y = 3(0)^2 + 12(0) + 15
\]

This simplifies to:

\[
y = 15
\]

### Conclusion:

The quadratic equation in standard form is:

\[
y = 3x^2 + 12x + 15
\]

The y-intercept is:

\[
(0, 15)
\]