Question

Question #1f2c9

Answer 1

a. Its really up to you but three possible pairs can be: `(2,3),(5,3),(10,3)`

b. Zero

c. (See Graph)

Explanation:

We can think of the equation/function of the line `y=3` to be `f(x)=3`. They mean the same exact thing but this notation of `f(x)` will help make sense of what I'm about to say next.

The notation `f(x)` basically means the value of the function f when evaluated at a number x.

So if I wanted to find the value of the function when `x=2` then I would simply substitute `2` for every `x` I see in the function:

Thus, `f(2)=3`

This is a very special case however which I will explain further in detail later but in English, this says, the value of the function when `x` is 2 is 3.

Interestingly this will be true for any value of `x`.

a. An ordered pair is a fancy way to say point. We are asked to find at least three points that satisfy the equation which is to say find three points on the graph of `y=3`

Remember how earlier I mentioned the notation `f(x)` basically means the value of the function f when evaluated at a number x. Another way to think about this is as an input and output relationship.

Similarly then, if I input `2` into the function, the output will be `3` by the same reasons of the first example and again, this will be true for any value of `x` in this particular case.

We can describe this input/output relationship graphically as an `(x,y)` coordinate.

Using the example above, I can thus translate `f(2)=3` to `(2,3)` which I can then plot on the graph.

We repeat this process for as many times as we want but remember that in this particular case, the output/value of the function will always be `3` so really any number will satisfy the equation.

So three coordinates/ordered pairs can be

`(2,3)`

`(5,3)`

`(10,3)`

But generally, `(x,3)` where `x` is any number.

b. The slope of `y=3` is the slope of a horizontal line which has a slope of ZERO and that is a given fact.

However, if you need to prove it then we can take two coordinates that we already found from above we can apply the slope formula:

`m=(y_2-y_1)/(x_2-x_1)`

Where one coordinate is denoted as `(x_1,y_1)` and the other `(x_2,y_2)`

By applying this formula, you should always get `0` as the slope no matter what two coordinates you choose.

c. Thinking generally, the line of `y=`anything is one that crosses the `y-`axis and is a horizontal line. There are multiple ways to graph this equation:

One way (and probably the easiest in this case) is to use the coordinates that you chose and plot them then draw a line through those lines. What you should end with is a straight horizontal line.

Another way to think about it is to look at the generally `y=mx+b` formula which the general equation of a line where `m` is the slope and `b` is the `y-`intercept. We have already established that the slope of this line is `0` and so looking at the equation `y=3`, it is really the equation `y=0x+3` which means the `y-`intercept is `3` Given that, we plot the `y-`intercept `(0,3)` and draw a horizontal line.

In the end the graph should look something like this:

graph{0x+3 [-5.136, 4.726, -0.157, 4.774]}