A function is a set of rules so that for every input, we get only one output. We represent functions in math as equations with two variables: x and y. The x is the input and y is the output. For every x we plug into the equation, we only get one y.

We can represent functions on a graph using the coordinate plane with our coordinates being x (the input) and y (the output) in the same way as we graphed things before, with our point (x, y). The functions we’ll see in the videos create a pattern of points that make a line.

Video Source (04:56 mins) | Transcript

In the previous video, we plotted some points from the function and then connected those points with a line. We do this when we have continuous data. If we were to choose any number between the points we plotted, the equation’s output would be on the line because of the pattern the function makes.

Sometimes we plot points that don’t make a perfect pattern or that have no values in between. This is called discrete data and we don’t connect the dots with a line. They just stay as a scatter plot.

Video Source (04:40 mins) | Transcript

A discrete graph is one with scattered points. They may or may not show a direction or trend. They don’t have data in between the points already given.

A continuous graph has a line because there is data in between the points already given.

Additional Resources

• Khan Academy: Graphing a Linear Equation: y = 2x + 7 (13:09 mins, Transcript)

Practice Problems

1. Is the graph below discrete or continuous?

(Solution

x

Solution:
Discrete
Details:
This shows discrete data because each point is distinct and not connected to one another.

)

2. Is the graph below discrete or continuous? (

Solution

x

Solution:
Continuous
Details:
This shows discrete data because each point is distinct and not connected to one another.

)

3. A chicken breeder sells pullets (young hens that have just started laying eggs) for \($25\) each. The breeder charges an additional \($100\) to deliver the animals, regardless of the number purchased. If we let P represent the number of pullets sold, then we can represent the total cost (T) as \(T = 100 + 25P\). Create a graph that illustrates the relationship between P and T, assuming that the number of pullets purchased is between 1 and 8.

(Hint: Use the equation for the total cost. Create a table with input values from 1 to 8, and calculate the outputs to find the coordinates that you need to graph.) (

Solution

x

Solution:


We know that the total cost of buying the pullets can be represented by the equation \(T = 100 + 25 P\), where T is the total cost and P is the number of pullets we have purchased.

If we purchase \({\color{Red}1}\) pullet, then we can find out the total cost by substituting 1 into the equation, \(T = 100 + 25 P\), for P:

\(T = 100 + 25 ({\color{Red}1})\)

Then multiply 25 by \({\color{Red}1}\) which gives us:

\(T = 100 + {\color{Red}25}\)

Then add 25 to 100:

\(T = 125\)

So the total cost for one pullet would be \($125\).

We continue the same process for purchasing 2-8 pullets.

P	T	How do we know the total cost?
1	125	T = 100 + 25P T = 100 + 25 (1) T = 100 + 25 T = 125
2	150	T = 100 + 25P T = 100 + 25 (2) T = 100 + 50 T = 150
3	175	T = 100 + 25P T = 100 + 25 (3) T = 100 + 75 T = 175
4	200	T = 100 + 25P T = 100 + 25 (4) T = 100 + 100 T = 200
5	225	T = 100 + 25P T = 100 + 25 (5) T = 100 + 125 T = 225
6	250	T = 100 + 25P T = 100 + 25 (6) T = 100 + 150 T = 250
7	275	T = 100 + 25P T = 100 + 25 (7) T = 100 + 175 T = 275
8	300	T = 100 + 25P T = 100 + 25 (8) T = 100 + 200 T = 300

Now we can plot each of the eight points on the graph:

)

4. Suppose that you are driving to a destination 100 kilometers away at a constant speed of R kilometers per hour (km/h). The amount of time required to reach your destination is \(\frac{100}{\text{R}}\). Choose values for R, and create a graph showing the relationship between the speed (R) and the time (T). (

Solution

x

Solution:
This graph shows the time required when the speed is between 30 and 130 km/h.

)

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