Question

the table shows the weekly profit in dollars earned from the sale of pastries at seven different prices. the data can be modeled by a quadratic function. write a function that best models this data.
x: 2.25,2.50,2.75,3.00,3.25,3.50,3.75
y: 149.75,166,174.75,176,169.75,156,134.75

Answer 1

To model the weekly profit \(y\) earned from the sale of pastries at different prices \(x\), we can use a quadratic function of the form:

\[
y = ax^2 + bx + c
\]

Given the data points:

\[
\begin{align*}
(2.25, 149.75) \\
(2.50, 166.00) \\
(2.75, 174.75) \\
(3.00, 176.00) \\
(3.25, 169.75) \\
(3.50, 156.00) \\
(3.75, 134.75) \\
\end{align*}
\]

We will use any three points to form a system of equations in order to find the coefficients \(a\), \(b\), and \(c\).

Let’s use the points \((2.50, 166)\), \((3.00, 176)\), and \((3.50, 156)\).

1. For \((2.50, 166)\):
\[
166 = a(2.50)^2 + b(2.50) + c
\]
\[
166 = 6.25a + 2.50b + c \quad \text{(Equation 1)}
\]

2. For \((3.00, 176)\):
\[
176 = a(3.00)^2 + b(3.00) + c
\]
\[
176 = 9a + 3b + c \quad \text{(Equation 2)}
\]

3. For \((3.50, 156)\):
\[
156 = a(3.50)^2 + b(3.50) + c
\]
\[
156 = 12.25a + 3.50b + c \quad \text{(Equation 3)}
\]

Now we have a system of equations:

\[
\begin{align*}
6.25a + 2.50b + c &= 166 \quad \text{(1)} \\
9a + 3b + c &= 176 \quad \text{(2)} \\
12.25a + 3.50b + c &= 156 \quad \text{(3)}
\end{align*}
\]

We can solve this system step by step:

**Step 1:** Subtract Equation 1 from Equation 2:

\[
(9a + 3b + c) - (6.25a + 2.50b + c) = 176 - 166
\]
\[
2.75a + 0.50b = 10 \quad \text{(Equation 4)}
\]

**Step 2:** Subtract Equation 2 from Equation 3:

\[
(12.25a + 3.50b + c) - (9a + 3b + c) = 156 - 176
\]
\[
3.25a + 0.50b = -20 \quad \text{(Equation 5)}
\]

**Step 3:** Now we can subtract Equation 4 from Equation 5:

\[
(3.25a + 0.50b) - (2.75a + 0.50b) = -20 - 10
\]
\[
0.50a = -30
\]
\[
a = -60
\]

**Step 4:** Substitute \(a\) back into Equation 4 to find \(b\):

\[
2.75(-60) + 0.50b = 10
\]
\[
-165 + 0.50b = 10
\]
\[
0.50b = 175
\]
\[
b = 350
\]

**Step 5:** Substitute \(a\) and \(b\) back into Equation 1 to find \(c\):

\[
6.25(-60) + 2.50(350) + c = 166
\]
\[
-375 + 875 + c = 166
\]
\[
c = 166 - 500
\]
\[
c = -334
\]

Now we've found \(a\), \(b\), and \(c\):

\[
a = -60, \quad b = 350, \quad c = -334
\]

Thus, the quadratic function that models the data is:

\[
y = -60x^2 + 350x - 334
\]