𝑦 = −3𝑥
2 − 12𝑥 + 8
a. complete the square and turn into vertex form
b. find vertex
c. find axis of symmetry
d. find 2 other points to plot
e. shift the equation 2 units to the left and 1 unit up

We are given the quadratic equation:

\[
y = -3x^2 - 12x + 8
\]

### Part a. Complete the square to turn it into vertex form.

1. **Factor out the coefficient of \( x^2 \):**
   We first factor out the \(-3\) from the terms involving \(x\):

   \[
   y = -3(x^2 + 4x) + 8
   \]

2. **Complete the square:**
   - To complete the square, take half of the coefficient of \(x\) (which is 4), square it, and add it inside the parentheses:
   \[
   \left( \frac{4}{2} \right)^2 = 4
   \]
   - Add and subtract 4 inside the parentheses:
   \[
   y = -3(x^2 + 4x + 4 - 4) + 8
   \]
   - This simplifies to:
   \[
   y = -3((x + 2)^2 - 4) + 8
   \]

3. **Simplify:**
   - Distribute the \(-3\):
   \[
   y = -3(x + 2)^2 + 12 + 8
   \]
   - Combine like terms:
   \[
   y = -3(x + 2)^2 + 20
   \]

Thus, the vertex form of the equation is:
\[
y = -3(x + 2)^2 + 20
\]

---

### Part b. Find the vertex.

From the vertex form \( y = -3(x + 2)^2 + 20 \), we know that the vertex is at the point:

\[
\text{Vertex} = (-2, 20)
\]

---

### Part c. Find the axis of symmetry.

The axis of symmetry for a quadratic equation in vertex form \( y = a(x - h)^2 + k \) is given by \( x = h \).

In this case, since \( h = -2 \), the axis of symmetry is:
\[
x = -2
\]

---

### Part d. Find two other points to plot.

We can find other points by plugging in values for \(x\) into the vertex form of the equation \( y = -3(x + 2)^2 + 20 \).

1. Let \( x = -1 \):
   \[
   y = -3(-1 + 2)^2 + 20 = -3(1)^2 + 20 = -3 + 20 = 17
   \]
   So one point is \( (-1, 17) \).

2. Let \( x = 0 \):
   \[
   y = -3(0 + 2)^2 + 20 = -3(2)^2 + 20 = -3(4) + 20 = -12 + 20 = 8
   \]
   So another point is \( (0, 8) \).

Thus, the two other points are \( (-1, 17) \) and \( (0, 8) \).

---

### Part e. Shift the equation 2 units to the left and 1 unit up.

To shift the equation \( y = -3(x + 2)^2 + 20 \) by 2 units to the left and 1 unit up:

1. Shift left by 2 units: Replace \( x \) with \( x + 2 \):
   \[
   y = -3((x + 2) + 2)^2 + 20 = -3(x + 4)^2 + 20
   \]

2. Shift up by 1 unit: Add 1 to the entire equation:
   \[
   y = -3(x + 4)^2 + 20 + 1 = -3(x + 4)^2 + 21
   \]

Thus, the new equation is:
\[
y = -3(x + 4)^2 + 21
\]

---

Do you have any questions or would you like more details on any of these steps?

Here are 5 related questions you can explore:
1. How do you convert from vertex form back to standard form?
2. What is the significance of the coefficient in front of \( (x + h)^2 \)?
3. How does completing the square help in solving quadratic equations?
4. Can you determine the range and domain of this quadratic function?
5. How does the shape of the graph change with different shifts and stretches?

**Tip:** The axis of symmetry always passes through the vertex of a parabola, making it a key point for analyzing its graph.

𝑦 = −3𝑥^2 − 12𝑥 + 8 a. complete the square and turn into vertex form b. find vertex c. find axis of symmetry d. find 2 other points to plot e. shift the equation 2 units to the left and 1 unit up

Learn how to complete the square for the quadratic equation y = −3x^2 − 12x + 8 to find its vertex form. This problem also covers finding the vertex, axis of symmetry, plotting additional points, and shifting the equation. Suitable for students in Grades 9-10.

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