Answer :
To determine the equation of the quadratic function that fits the given points in the table, follow these steps:
1. Form the general quadratic equation:
The general form of a quadratic equation is \( y = ax^2 + bx + c \).
2. Set up equations using the given points:
For each point \((x, y)\) in the table, substitute \(x\) and \(y\) into the general quadratic equation.
Given points:
- When \( x = -3 \), \( y = 3.75 \):
[tex]\[ 3.75 = a(-3)^2 + b(-3) + c \implies 3.75 = 9a - 3b + c \][/tex]
- When \( x = -2 \), \( y = 4 \):
[tex]\[ 4 = a(-2)^2 + b(-2) + c \implies 4 = 4a - 2b + c \][/tex]
- When \( x = -1 \), \( y = 3.75 \):
[tex]\[ 3.75 = a(-1)^2 + b(-1) + c \implies 3.75 = a - b + c \][/tex]
3. Solve the system of equations:
[tex]\[ \begin{cases} 9a - 3b + c = 3.75 \\ 4a - 2b + c = 4 \\ a - b + c = 3.75 \\ \end{cases} \][/tex]
Upon solving these equations simultaneously, we get:
- \( a = -0.25 \)
- \( b = -1 \)
- \( c = 3 \)
4. Form the quadratic equation:
Substitute \(a\), \(b\), and \(c\) back into the general form:
[tex]\[ y = -0.25x^2 - x + 3 \][/tex]
Therefore, the correct equation of the quadratic function represented by the table is:
[tex]\[ y = -0.25x^2 - x + 3 \][/tex]
In the given format:
[tex]\(\boxed{y = -0.25x^2 - x + 3}\)[/tex]
1. Form the general quadratic equation:
The general form of a quadratic equation is \( y = ax^2 + bx + c \).
2. Set up equations using the given points:
For each point \((x, y)\) in the table, substitute \(x\) and \(y\) into the general quadratic equation.
Given points:
- When \( x = -3 \), \( y = 3.75 \):
[tex]\[ 3.75 = a(-3)^2 + b(-3) + c \implies 3.75 = 9a - 3b + c \][/tex]
- When \( x = -2 \), \( y = 4 \):
[tex]\[ 4 = a(-2)^2 + b(-2) + c \implies 4 = 4a - 2b + c \][/tex]
- When \( x = -1 \), \( y = 3.75 \):
[tex]\[ 3.75 = a(-1)^2 + b(-1) + c \implies 3.75 = a - b + c \][/tex]
3. Solve the system of equations:
[tex]\[ \begin{cases} 9a - 3b + c = 3.75 \\ 4a - 2b + c = 4 \\ a - b + c = 3.75 \\ \end{cases} \][/tex]
Upon solving these equations simultaneously, we get:
- \( a = -0.25 \)
- \( b = -1 \)
- \( c = 3 \)
4. Form the quadratic equation:
Substitute \(a\), \(b\), and \(c\) back into the general form:
[tex]\[ y = -0.25x^2 - x + 3 \][/tex]
Therefore, the correct equation of the quadratic function represented by the table is:
[tex]\[ y = -0.25x^2 - x + 3 \][/tex]
In the given format:
[tex]\(\boxed{y = -0.25x^2 - x + 3}\)[/tex]
