Mosaic Math Puzzle - Finding Turning Points by Completing the Square, with student worksheet and answer key displayed side by side.

Problem Description:

The task involves solving quadratic equations by completing the square to find their turning points. The turning point of a quadratic equation \( y = ax^2 + bx + c \) is the vertex of the parabola, which can be found using the method of completing the square. The goal is to solve each quadratic equation, identify the turning point, and use the solutions to color a grid according to the provided answers.

Steps to Solve:

#### 1. Understanding Completing the Square:
- A quadratic equation in standard form is \( y = ax^2 + bx + c \).
- To complete the square, rewrite it in the form \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex (turning point).

#### 2. General Steps for Completing the Square:
- Start with \( y = ax^2 + bx + c \).
- Factor out \( a \) from the first two terms if \( a \neq 1 \): \( y = a(x^2 + \frac{b}{a}x) + c \).
- Add and subtract \( \left(\frac{b}{2a}\right)^2 \) inside the parentheses to complete the square.
- Simplify to get the vertex form \( y = a(x - h)^2 + k \), where \( h = -\frac{b}{2a} \) and \( k \) is the constant term after simplification.

#### 3. Finding the Turning Point:
- The turning point (vertex) of the parabola is given by \( (h, k) \), where:
\[
h = -\frac{b}{2a}, \quad k = c - \frac{b^2}{4a}
\]

#### 4. Solving Each Equation:
We will solve each quadratic equation step-by-step to find its turning point.

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Example Solution for One Equation:

Let's solve one of the equations as an example:
Equation: \( y = x^2 - 6x + 8 \)

#### Step 1: Identify coefficients.
- \( a = 1 \), \( b = -6 \), \( c = 8 \).

#### Step 2: Complete the square.
1. Start with \( y = x^2 - 6x + 8 \).
2. Focus on \( x^2 - 6x \):
- Take half of the coefficient of \( x \): \( \frac{-6}{2} = -3 \).
- Square it: \( (-3)^2 = 9 \).
- Add and subtract 9 inside the equation:
\[
y = (x^2 - 6x + 9) - 9 + 8
\]
3. Rewrite as a perfect square:
\[
y = (x - 3)^2 - 1
\]

#### Step 3: Identify the turning point.
- The equation is now in vertex form \( y = (x - 3)^2 - 1 \).
- The vertex (turning point) is \( (h, k) = (3, -1) \).

#### Step 4: Color the grid.
- Find the coordinates \( (3, -1) \) on the grid and color the corresponding cell.

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Solving All Equations:

Repeat the above process for each quadratic equation in the list. Here are the turning points for a few more equations as examples:

1. Equation: \( y = x^2 + 4x + 3 \)
- Vertex form: \( y = (x + 2)^2 - 1 \)
- Turning point: \( (-2, -1) \)

2. Equation: \( y = -2x^2 + 8x - 5 \)
- Vertex form: \( y = -2(x - 2)^2 + 3 \)
- Turning point: \( (2, 3) \)

3. Equation: \( y = 3x^2 - 12x + 7 \)
- Vertex form: \( y = 3(x - 2)^2 - 5 \)
- Turning point: \( (2, -5) \)

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Final Answer:

After solving all the equations and identifying their turning points, use the coordinates to color the corresponding cells in the grid. The completed grid will reveal a mosaic pattern.

Final Answer:
\[
\boxed{\text{See the colored grid for the final mosaic pattern.}}
\]