Question

a bakery is selling loaves of bread. the owner has observed that if they charge $2 or $10 per loaf, they make no profit.
the table below shows additional expected profits based on various prices for the loaf.

price per loaf ($x$)	profit ($p(x)$)
$4	$48
$6	$64
$8	$48
$10	$0

create a quadratic function of the form $p(x) = a(x - m)(x - n)$, where $a$, $m$, and $n$ are integers, that models the bakery’s profit $p(x)$ in terms of the price $x$ per loaf.

Explanation:

Step1: Identify roots from profit=0

We know that when \( P(x) = 0 \), \( x = 2 \) or \( x = 10 \). So the roots \( m = 2 \) and \( n = 10 \). Thus, the quadratic function is \( P(x)=a(x - 2)(x - 10) \).

Step2: Find 'a' using a point

We can use the point \( (4, 48) \) (when \( x = 4 \), \( P(x)=48 \)). Substitute into the function:
\( 48=a(4 - 2)(4 - 10) \)
\( 48=a(2)(- 6) \)
\( 48=-12a \)
Solve for \( a \): \( a=\frac{48}{-12}=- 4 \).

Step3: Write the final function

Substitute \( a=-4 \), \( m = 2 \), \( n = 10 \) into \( P(x)=a(x - m)(x - n) \):
\( P(x)=-4(x - 2)(x - 10) \)

Answer:

\( P(x)=-4(x - 2)(x - 10) \)