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The image shows a graph with several curves, each representing quadratic functions (degree 2 polynomials). The task asks to find the factored form of each quadratic function based on their graphs.

Here’s how we can approach it:

1. **Identify Roots of Each Curve:**
   The roots (x-intercepts) of each quadratic function are key to factorizing them. Quadratic functions in factored form take the shape \( f(x) = a(x - r_1)(x - r_2) \), where \( r_1 \) and \( r_2 \) are the roots.

2. **Analysis of the Curves:**
   - **Curve \( C_1 \) (Red)**: Appears to have roots at \( x = 0 \) and \( x = 1 \), and opens upwards.
   - **Curve \( C_2 \) (Green)**: Appears to have roots at \( x = 0 \) and \( x = 2 \), and opens downwards.
   - **Curve \( C_3 \) (Blue)**: Appears to have roots at \( x = -1 \) and \( x = 1 \), and opens upwards.
   - **Curve \( C_4 \) (Orange)**: Appears to have a double root at \( x = 0 \) (touches the x-axis at a single point), and opens upwards.

3. **Factored Forms Based on Roots:**
   - For \( C_1 \), the factored form is \( f(x) = a(x)(x - 1) \).
   - For \( C_2 \), the factored form is \( f(x) = a(x)(x - 2) \), with a negative \( a \) since the parabola opens downward.
   - For \( C_3 \), the factored form is \( f(x) = a(x + 1)(x - 1) \).
   - For \( C_4 \), the factored form is \( f(x) = a(x^2) \), as it has a double root at \( x = 0 \).

Now, we would need to check the orientation (whether the parabola opens upward or downward) and apply an appropriate value for \( a \), which can be inferred from the graph.

Would you like to proceed with calculating the exact forms, or would you like to clarify something?

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Here are some relative questions for deeper understanding:

1. How can you determine the value of \( a \) in the factored form based on the curve’s shape?
2. What role do the roots of the quadratic function play in determining its factored form?
3. How do you determine whether the parabola opens upwards or downwards from the graph?
4. How would you confirm the factored form by expanding and comparing it to the standard quadratic form?
5. What happens to the factored form when a quadratic function has a double root?

**Tip:** To factor any quadratic equation, always start by finding the x-intercepts or roots first, which will guide you in writing the factored form.

Retrouver la forme factorisée de chaque fonction à partir des courbes représentatives de fonctions polynômes de degré 2.

Discover how to find the factored forms of quadratic functions by analyzing their graphical representations. This problem covers essential algebraic concepts such as the zero product property and quadratic factorization. Ideal for high school students in Grades 10-12.

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