To find the equation of a quadratic function that passes through the points $(-3,-1)$, $(-2,-3)$, $(-4,5)$, $(-1,-1)$, and $(0,5)$, we can start by assuming the general form of a quadratic function:

$f(x) = ax^2 + bx + c$

We will use three points to find the coefficients $a$, $b$, and $c$, as a quadratic equation is determined by three points. Let's select three points: $(-3, -1)$, $(-2, -3)$, and $(-4, 5)$. Then we will substitute the coordinates into the equation $f(x) = ax^2 + bx + c$ to form a system of equations.

Step 1: Write down the equations

Using the point $(-3, -1)$: $a(-3)^2 + b(-3) + c = -1 \quad \Rightarrow \quad 9a - 3b + c = -1$

Using the point $(-2, -3)$: $a(-2)^2 + b(-2) + c = -3 \quad \Rightarrow \quad 4a - 2b + c = -3$

Using the point $(-4, 5)$: $a(-4)^2 + b(-4) + c = 5 \quad \Rightarrow \quad 16a - 4b + c = 5$

Step 2: Solve the system of equations

We now have the system of equations:

1. $9a - 3b + c = -1$
2. $4a - 2b + c = -3$
3. $16a - 4b + c = 5$

We can solve this system to find $a$, $b$, and $c$. Let me calculate that.The solution to the system of equations gives us the values:

$a = 2, \quad b = 8, \quad c = 5$

Thus, the equation of the quadratic function is:

$f(x) = 2x^2 + 8x + 5$

This quadratic function passes through the points $(-3, -1)$, $(-2, -3)$, and $(-4, 5)$. You can also verify that it satisfies the other two points $(-1, -1)$ and $(0, 5)$.

Would you like a detailed verification or have any questions?

Here are some related questions:

1. How do you verify if a point lies on a quadratic function?
2. What methods can be used to solve systems of equations like the one we solved?
3. How does changing $a$, $b$, and $c$ affect the graph of a quadratic function?
4. What is the significance of the vertex in a quadratic function?
5. How can you determine the axis of symmetry for any quadratic function?

Tip: To quickly verify if a point lies on a function, substitute the coordinates into the equation and check if both sides are equal.