(-3,-1)(-2,-3)(-4,5)(-1,-1)(0,5) find the equation of this quadratic function

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.

Discover how to find the equation of a quadratic function that passes through the points (-3, -1), (-2, -3), (-4, 5), (-1, -1), and (0, 5). This problem involves forming and solving a system of equations to determine the coefficients of the quadratic function. Suitable for students in Grades 9-11.

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