To find the equation of the quadratic function that fits the given table of values, we will start assuming the standard form of the quadratic function which is:
[tex]\[ y = a(x - b)^2 + c \][/tex]
We have the following points from the table:
1. [tex]\((-3, 3.75)\)[/tex]
2. [tex]\((-2, 4)\)[/tex]
3. [tex]\((-1, 3.75)\)[/tex]
4. [tex]\((0, 3)\)[/tex]
5. [tex]\((1, 1.75)\)[/tex]
We need to find the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex].
### Step 1: Identify the Vertex
From the table, notice that the y-values increase and then decrease, indicating a parabola which opens upwards or downwards. The vertex of the parabola occurs at the maximum or minimum point of the function.
From the table:
- At [tex]\(x = -2\)[/tex], [tex]\(y\)[/tex] is maximum (since it's not the maximum in the table but it's higher than its neighbors).
Thus, the vertex is at [tex]\((-2, 4)\)[/tex].
So, [tex]\(b = -2\)[/tex] and [tex]\(c = 4\)[/tex].
### Step 2: Substitute the Vertex Form
We now have part of the quadratic equation in the form:
[tex]\[ y = a(x - (-2))^2 + 4 \][/tex]
[tex]\[ y = a(x + 2)^2 + 4 \][/tex]
### Step 3: Use Another Point to Solve for [tex]\(a\)[/tex]
We can use any other point to find the value of [tex]\(a\)[/tex]. Let's use the point [tex]\((-3, 3.75)\)[/tex].
Substitute [tex]\((-3, 3.75)\)[/tex] into the equation:
[tex]\[ 3.75 = a(-3 + 2)^2 + 4 \][/tex]
[tex]\[ 3.75 = a(-1)^2 + 4 \][/tex]
[tex]\[ 3.75 = a(1) + 4 \][/tex]
[tex]\[ 3.75 = a + 4 \][/tex]
Solving for [tex]\(a\)[/tex]:
[tex]\[ a = 3.75 - 4 \][/tex]
[tex]\[ a = -0.25 \][/tex]
### Step 4: Verify using Another Point
We'll verify by using another point [tex]\((0, 3)\)[/tex]:
[tex]\[ y = -0.25(x + 2)^2 + 4 \][/tex]
Substitute [tex]\(x = 0\)[/tex]:
[tex]\[ 3 = -0.25(0 + 2)^2 + 4 \][/tex]
[tex]\[ 3 = -0.25(2)^2 + 4 \][/tex]
[tex]\[ 3 = -0.25(4) + 4 \][/tex]
[tex]\[ 3 = -1 + 4 \][/tex]
[tex]\[ 3 = 3 \][/tex]
The equation verifies correctly.
### Final Quadratic Equation
The quadratic function that represents the given table is:
[tex]\[ y = -0.25(x + 2)^2 + 4 \][/tex]
In the format [tex]\(y = a(x - b)^2 + c\)[/tex], the values are:
[tex]\[ a = -0.25 \][/tex]
[tex]\[ b = -2 \][/tex]
[tex]\[ c = 4 \][/tex]
So the complete quadratic equation is:
[tex]\[ y = -0.25(x + 2)^2 + 4 \][/tex]
