To solve this problem, let's analyze each quadratic function provided and match it with its corresponding graph by identifying key features, particularly the y-intercept and the direction of the parabola.

Step 1: Identify Functions

The given functions are:

1. $f(x) = x^2 + 8x + 15$
2. $f(x) = x^2 - 8x + 15$
3. $f(x) = x^2 + 2x - 15$
4. $f(x) = x^2 - 2x - 15$

Step 2: Analyze Each Function

Each function is a quadratic equation in the form $f(x) = ax^2 + bx + c$. Key features we can derive from these are:

• Y-intercept: This is the constant term $c$ in the function.
• Vertex and Direction: Since all of these have $a = 1$, they open upwards. We can complete the square or use the formula for the vertex to find the turning points, or observe transformations.

1. $f(x) = x^2 + 8x + 15$

   • Y-intercept: $15$
   • Vertex: Completing the square or using the vertex formula $x = -\frac{b}{2a}$, we find $x = -4$.
   • Substitute $x = -4$ back into the function to find the minimum y-value (vertex).

2. $f(x) = x^2 - 8x + 15$

   • Y-intercept: $15$
   • Vertex: $x = 4$
   • Substitute $x = 4$ to find the vertex.

3. $f(x) = x^2 + 2x - 15$

   • Y-intercept: $-15$
   • Vertex: $x = -1$

4. $f(x) = x^2 - 2x - 15$

   • Y-intercept: $-15$
   • Vertex: $x = 1$

Step 3: Match to Graphs

Using the information derived, we can then match each function to its graph based on the vertex, y-intercept, and shape.

Would you like a breakdown of the calculations for each function's vertex, or do you have any specific questions on how to proceed?

Here are 5 related questions that build on this topic:

1. How can you determine the vertex of a quadratic function without completing the square?
2. What does the coefficient of $x^2$ tell us about the shape and direction of a parabola?
3. How does changing the constant term in a quadratic function affect its graph?
4. How can you quickly identify the y-intercept from the standard form of a quadratic equation?
5. How does the sign of the coefficient of $x$ affect the position of the vertex along the x-axis?

Tip: Remember, for any quadratic function $f(x) = ax^2 + bx + c$, the vertex x-coordinate is always given by $x = -\frac{b}{2a}$.