Question

multiple choice 1 point
a small business is selling custom t-shirt. the business owner has observed that if they charge $40 or $60 per shirt, they make no profit.

price per shirt (x) | profit p(x)
$40 | 0
$45 | 100
$50 | 250
$55 | 100
$60 | 0

the table above shows additional expected profits based on various prices for the shirt.
create a quadratic function of the form $p(x)=a(x-m)(x-n)$, where a, m, and n are integers, that models the businesss profit p(x) in terms of the price x per shirt.

$p(x)=2(x-40)(x-60)$
$p(x)=-2(x-40)(x-60)$
$p(x)=(x-40)(x-60)$
$p(x)=-(x-40)(x-60)$
clear my selection

Explanation:

Step1: Identify roots of function

The roots are $x=40$ and $x=60$, so $P(x)=a(x-40)(x-60)$

Step2: Substitute known point to find $a$

Use $x=50, P(x)=200$:
$200 = a(50-40)(50-60)$
$200 = a(10)(-10)$
$200 = -100a$

Step3: Solve for $a$

$a = \frac{200}{-100} = -2$

Step4: Write final function

$P(x) = -2(x - 40)(x - 60)$

Answer:

$P(x) = -2(x - 40)(x - 60)$