Unit 7, Lesson 7
Building Quadratic Functions to Describe Situations (Part 3)
Algebra 1
7.1
Warm-up
Which three go together? Why do they go together?
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Which three go together? Why do they go together?
A company that sells movies online is deciding how much to charge customers to download a new movie. Based on data from previous sales, the company predicts that if they charge dollars for each download, then the number of downloads, in thousands, is .
| price (dollars per download) | number of downloads (thousands) | revenue (thousands of dollars) |
|---|---|---|
| 3 | 15 | 45 |
| 5 | ||
| 10 | ||
| 12 | ||
| 15 | ||
| 18 | ||
Plot the points that represent the revenue, , as a function of the price of one download in dollars, .
Blank coordinate plane with grid, origin O. Horizontal axis scale 0 to 25 by 5’s, labeled “price (dollars per download)”. Vertical axis scale 0 to 200 by 50’s, labeled “revenue (thousands of dollars)”.
Here are four sets of descriptions and equations that represent some familiar quadratic functions. The graphs show what graphing technology may produce when the equations are graphed. For each function:
The area of a rectangle with a perimeter of 25 meters and a side length of :
Graph of the quadratic function
on a coordinate plane, origin
. Horizontal axis scale negative 10 to 15 by 5s, labeled “length (meters)”. Vertical axis scale 0 to 60 by 10’s, labeled “area (square meters)”. Some of the points of this function are
(0 comma 0), (1 comma 11 point 5) and (5 comma 37 point 5) to a maximum at (6 point 2 5 comma 39 point 0 6 3) then decreasing through (10 comma 25), (12 comma 6) and (12 point 5 comma 0).
Domain:
Vertex:
Zeros:
The number of squares as a function of step number :
Graph of the quadratic function
on a coordinate plane, origin
. Horizontal axis scale negative 12 to 12 by 6’s, labeled “step number”. Vertical axis scale 0 to 200 by 40’s, labeled “number of squares”. Some of the points of this function are (negative 10 comma 104), (negative 6 comma 40) and (negative 3 comma 13) to a minimum at (0 comma 4) then increasing through (3 comma 13), (6 comma 40) and (10 comma 104).
Domain:
Vertex:
Zeros:
The distance, in feet, that an object has fallen seconds after being dropped:
Graph of the quadratic function
on a coordinate plane, origin
. Horizontal axis scale negative 8 to 8 by 4’s, labeled “time (seconds)”. Vertical axis scale 0 to 600 by 200’s, labeled “distance fallen (feet)”. Some of the points of this function are (negative 8 comma 1024), (negative 5 comma 400) and (negative 1 comma 16) to a minimum at
(0 comma 0) then increasing through (1 comma 16), (5 comma 400) and (8 comma 1024).
Domain:
Vertex:
Zeros:
The height, in feet, of an object seconds after being dropped:
Graph of the quadratic function h(t)=576 - 16t^2 on a coordinate plane, origin O. Horizontal axis scale negative 8 to 8 by 4’s, labeled “time (seconds)”. Vertical axis scale 0 to 800 by 200’s, labeled “height (feet)”. Some of the points of this function are (negative 6 comma 0), (negative 4 comma 320) and (negative 2 comma 512) to a maximum at (0 comma 576) then decreasing through (2 comma 512), (4 comma 320) and (6 comma 0).
Domain:
Vertex:
Zeros:
Quadratic functions often come up when studying revenue, which is the amount of money collected when selling something.
Suppose we are selling raffle tickets and deciding how much to charge for each ticket. When the price of the tickets is higher, typically fewer tickets will be sold.
Let’s say that with a price of dollars, it is possible to sell tickets. We can find the revenue by multiplying the price by the number of tickets expected to be sold. A function that models the revenue, , collected is . Here is a graph that represents the function.
Graph of non linear function, origin O. Horizontal axis, price, dollars, from 0 to 9 by 1’s. Vertical axis, revenue, thousands of dollars, 0 to 1400 by 200’s. Line starts at 0 comma 0, increases until 4 comma 1200 then decreases until 8 comma 0. Passes through 2 comma 900 and 6 comma 900.
When ticket prices are low, a lot of tickets may be sold, but the total revenue is still low because the tickets are cheap. When the ticket prices get close to \$8, not many tickets are sold so the revenue is low again. From the graph, we can tell that the greatest revenue comes when there is a balance between ticket price and number of tickets sold. In this situation, that is \$1,200 of revenue when tickets are sold for \$4 each.
We can also see that for function , the domain is between 0 and 8. This makes sense because the cost of the tickets can’t be negative. If the price is more than \$8, the model doesn’t work because the revenue collected can’t be negative. A negative revenue (based on a non-negative ticket price) could occur only if the number of tickets sold is negative, which is not possible.