To match each quadratic graph with its respective function, let's analyze the key characteristics of each graph and then compare them to the given functions.

Steps to Match the Graphs

1. Identify the Roots: Each function is given in factored form, $f(x) = a(x - r_1)(x - r_2)$, where $r_1$ and $r_2$ are the roots (x-intercepts) of the function. Look at each graph to determine its x-intercepts and then find which function has those corresponding roots.

2. Determine the Direction of Opening:

   • If the coefficient $a$ in front of the function is negative, the parabola opens downward.
   • If $a$ is positive, the parabola opens upward.

3. Look for Vertical Stretch or Compression:

   • A coefficient $a > 1$ (e.g., 2) will make the parabola narrower.
   • A coefficient $0 < a < 1$ (e.g., 0.5) will make it wider.

Analyzing the Functions

Given functions:

1. $f(x) = -2(x - 1)(x + 3)$ - Opens downward with roots at $x = 1$ and $x = -3$.
2. $f(x) = (x - 3)(x + 1)$ - Opens upward with roots at $x = 3$ and $x = -1$.
3. $f(x) = 0.5(x - 6)(x + 2)$ - Opens upward with roots at $x = 6$ and $x = -2$, and is wider than a standard parabola.
4. $f(x) = 2(x + 6)(x - 2)$ - Opens upward with roots at $x = -6$ and $x = 2$, and is narrower than a standard parabola.
5. $f(x) = (x + 3)(x - 1)$ - Opens upward with roots at $x = -3$ and $x = 1$.
6. $f(x) = 0.5(x + 6)(x - 2)$ - Opens upward with roots at $x = -6$ and $x = 2$, and is wider than a standard parabola.

Analyzing the Graphs

1. First Graph:

   • Opens downward.
   • Roots are approximately $x = -3$ and $x = 1$.
   • Matches $f(x) = -2(x - 1)(x + 3)$.

2. Second Graph:

   • Opens upward.
   • Roots are approximately $x = -6$ and $x = 2$.
   • Wide shape, indicating a coefficient less than 1.
   • Matches $f(x) = 0.5(x + 6)(x - 2)$.

3. Third Graph:

   • Opens upward.
   • Roots are approximately $x = -3$ and $x = 1$.
   • Matches $f(x) = (x + 3)(x - 1)$.

Conclusion

The matches are:

• First Graph: $f(x) = -2(x - 1)(x + 3)$
• Second Graph: $f(x) = 0.5(x + 6)(x - 2)$
• Third Graph: $f(x) = (x + 3)(x - 1)$

Would you like more details on the solution process or further clarification?

Expanded Questions:

1. What are the general steps to identify the characteristics of a quadratic function from its graph?
2. How does the value of the coefficient $a$ affect the shape of a parabola?
3. What role do the roots play in determining the function of a parabola?
4. How can you determine if a quadratic opens up or down just by looking at the equation?
5. How does vertical compression/stretch relate to the value of $a$?

Tip:

When matching functions to graphs, start by identifying roots and the direction of opening, as they provide immediate clues to the correct function.