Question 567·Hard·Algebra

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A small business’s profit $P$ (in dollars) is a linear function of the number of units sold, $u$ . The business breaks even when 150 units are sold, and earns $3,300 profit when 300 units are sold.

Which equation expresses $P$ as a function of $u$ ?

A linear function changes by the same amount for each extra unit sold. Breaking even at a stated sales count gives a zero-profit point, not the profit at zero sales. Pair that point with the other profit figure, then fit a line to both points in a Desmos table. Don't mistake a profit earned after sales for the starting profit.

Hints

Hint 1
A break-even point has zero profit. Which number of units is paired with that zero?
Hint 2
Each sales count and its profit form an input-output pair. Put both pairs in a Desmos table so the same line must fit them.
Hint 3
In a linear model, the intercept is the profit at zero units sold. Neither stated profit occurs at zero units, so let the two points determine it.

Step-by-step

Approach 1: Fit a line through the two profit points
Step 1
Translate break-even into a point
Break even means the profit is zero. So $P(150)=0$ , giving the point $(150,0)$ , where the first number is units sold and the second is profit. Break-even means zero profit at the stated sales count, not zero units sold.
Step 2
Translate the other profit into a point
The business earns $3,300 when it sells $300$ units, so $P(300)=3300$ . That gives the second point, $(300,3300)$ . Because the inputs differ, these two points determine one line.
Step 3
Find the rate and starting profit in Desmos
Enter the unit counts as $x_1$ and the profits as $y_1$ in a table. Type $y_1\sim mx_1+b$ ; the $\sim$ tells Desmos to fit one line to both rows. Under PARAMETERS, Desmos shows $m=22$ and $b=-3300$ . The slope $m$ is the profit change per extra unit; the intercept $b$ is profit at zero units.
Step 4
Write profit as a function of units
Use the fitted values in $P=mu+b$ :
$P=22u-3300.$
The minus sign matters: at zero units sold, the model gives a $3,300 loss, not a $3,300 profit. Choice D.

Approach 2: Build the rule from break-even
Step 1
Find the profit gained per extra unit
From $150$ to $300$ units sold, profit rises from $0$ to $3300$ . Type $\frac{3300-0}{300-150}$ in Desmos. It prints $22$ , so the rate of change is $22 of profit per extra unit.
Step 2
Anchor profit at break-even
The expression $u-150$ counts units sold past break-even. Each one adds $22 to the zero profit at $150$ units, so $P=22(u-150)$ . This is point-slope form: it starts at a known point and adds the rate times the change in input.
Step 3
Match the rule to an equation
Distribute $22$ to put the rule in the form shown by the choices:
$P=22(u-150)$ $P=22u-3300.$
So profit increases by $22 per unit and is zero at 150 units. Choice D.

Question 567·Hard·Algebra

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