The image shows a quadratic function's graph and provides multiple options for equations. The key strategy to figure out which equations match the graph involves analyzing the graph's features, such as:

1. Vertex: The graph's vertex appears to be at the point (-5, 4). This is a crucial piece of information since a quadratic equation can be written in vertex form:
   $f(x) = a(x - h)^2 + k$
   where $(h, k)$ is the vertex.

2. Symmetry and shape: The graph is symmetric, and the parabola opens downward, which means the coefficient "a" in the equation should be negative.

3. Points on the graph: You can check specific points from the graph, such as (-7, 0) and (-3, 0), which correspond to the x-intercepts. This can help confirm factors of the equation.

Now, let's review the provided equations:

• A: $f(x) = -x^2 - 10x - 21$
• B: $f(x) = x^2 + 10x + 21$
• C: $f(x) = -(x + 5)^2 + 4$
• D: $f(x) = (x + 5)^2 - 4$
• E: $f(x) = -(x + 3)(x + 7)$
• F: $f(x) = -(x - 3)(x - 7)$

Let's go step by step:

• Vertex Form: From the graph, the vertex is at (-5, 4). This matches equation C, which is in vertex form with the vertex $(-5, 4)$, and the negative sign indicates the parabola opens downward.

• Intercepts and Factoring: The x-intercepts are at -7 and -3, so the equation should factor as $(x + 7)(x + 3)$. Thus, equation E is a good match because it has these intercepts and a negative sign for the downward-opening parabola.

Therefore, the correct equations are likely C and E.

Strategy:

1. Identify the vertex.
2. Check if the parabola opens upward or downward (negative coefficient).
3. Look for specific points, such as x-intercepts, to match factored forms.

Do you need further explanations or details on any specific part?

Related Questions:

1. What does the vertex form of a quadratic function tell us about the graph?
2. How can you use the x-intercepts to factor a quadratic equation?
3. What is the difference between standard and vertex form of quadratic equations?
4. How do you determine the direction in which a parabola opens?
5. What is the role of the "a" value in a quadratic equation?

Tip: Always start by identifying the vertex and direction of the parabola to narrow down possible equation choices.