At the start of this module, you were wondering whether you could earn a profit by making bikes and were given a profit function.
[latex]x[/latex] = the number of bikes produced and sold
[latex]P(x)[/latex] = profit as a function of x
[latex]R(x)[/latex] = revenue as a function of x
[latex]C(x)[/latex] = cost as a function of x
Profit, revenue, and cost are all linear functions. They are a function of the number of bikes sold. Remember that you were planning to sell each bike for $600 and it cost you $1,600 for fixed costs plus $200 per bike.
Since you take in $600 for each bike you sell, revenue is:
Your costs are $200 per bike plus a fixed cost of $1600, so your overall cost is:
That means that profit becomes:
With that in mind, let’s take another look at the table. The profit is found by subtracting the cost function from the revenue function for each number of bikes.
|Number of bikes ([latex]x[/latex])||Profit ($)|
That still doesn’t explain how to figure out the break-even point, the number of bikes for which the revenue equals the costs.
One way is to set the functions equal to one another and solve for [latex]x[/latex].
That means that if you sell 4 bikes, you will take in the same amount you spent to make the bikes. So selling less than 4 bikes will result in a loss and selling more than 4 bikes will result in a profit.
Another method for finding the break-even point is by graphing the two functions. You can use whichever method of graphing you find useful.
To plot points, for example, make a list of values for each function. Then use them to determine coordinates that you can plot.
To use the slope and y-intercept method, first find the y-intercept by setting x = 0. Then determine the slope as the coefficient of the variable, which is the m value.
|y-intercept = 0||y-intercept = 1600|
|slope = 600||slope = 200|
Using this information, the functions can be graphed
Now we can see the break-even point right away. It is the point at which the cost function intersects the revenue function. It occurs when [latex]x=4[/latex]. At this point you neither make a profit nor incur a loss.
The last question to consider is how changing your price affects your profit. If your costs remain the same, increasing the price you charge will shift your break-even point to a lower number of bikes and increase your revenue for every value of [latex]x[/latex]. However, people may not buy as many bikes. Lowering your price will shift your break-even point to a higher number of bikes and decrease your revenue for every value of [latex]x[/latex]. However, more people may buy your bikes.
Let’s take a look at the graphs for two other possible prices.
Graphing lets you quickly see a visual representation of the functions and how they are related to one another. Notice how the break-even point shifts to 8 bikes for a lower price and only 2 bikes for a higher price.
So writing, graphing, and comparing linear functions can be quite useful. As for deciding what price people are willing to pay for your bike, that’s a whole different topic!