Expert: Josh - 1/28/2005

Question
1. Can every line be written in slope-intercept form? Explain.
2. Does every line have two distinct intercepts (for example, one x-intercept and one yintercept)?
Explain.
3. Are there any lines that have no intercepts at all? Explain.
4. What can you say about two lines that have equal slopes and equal y-intercepts?
5. What can you say about two lines with the same x-intercept and the same y-intercept?
6. If two lines have the same slope but different x-intercepts, can they have the same yintercept?
7. If two lines have the same y-intercept but different slopes, can they have the same xintercept?
8. How may x-intercepts can the graph of a function have? How many y-intercepts?
9. Is a graph that consists of just one data point a function? Can your write an equation for
this function, if it exists?
10. Is there a function whose graph is symmetric with respect to the y-axis? To the x-axis?

Answer
Gina,

You really should attempt the problem set or at least have a go at answering the questions before you ask for help.
It would be useful if you actually indicate what aspect of the questions that you don't understand.

A1. Yes. A straight line is completely characterized by its slope and a given point on the line. In particular, given its slope and an intersecting point with either the x-axis or y-axis, we can uniquely identify the line. In the intercept form y=mx+b, m represents the gradient (or slope) and b represents the y-intercept. So, it captures all the attributes of a straight line.

A2. Provided the gradient (or slope) is not 0 or infinity.
A slope of zero corresponds to a horizontal line. A slope of infinity corresponds to a vertical line. They only intersect with one axis.

A3. No. Unless you reduce a straight line to a single point.

A4. This follows from part 1. It means that the 2 lines are colinear and identical.

A5. Again, they are the same. The reason is because you started out with a point that they have in common and the second point in common must force them both to have the same slope.

A6. No, I suggest that you draw a diagram to see this (simply by sliding the ruler).
They must be parallel.

A7. No.

A8. Not sure if you have really covered this. The answer depends on the degree of the polynomial. With straight lines, you are dealing with a polynomial of degree 1. With quadratic equations, you are dealing with a polynomial of degree 2, and there can be at most 2 intersecting points with the x-axis. With cubic equations, you are dealing with a polynomial of degree 3, and there can be at most 3 intersecting points with the x-axis.
A function is a general concept, it simply tells us how a variable (say y or f(x)) depends on another variable (say, x, or powers of x and so forth).

A9. This is getting a bit pedantic. If you are talking about a continuous function, it must be well defined within some sort of interval. If you are talking about a discrete function (instead of drawing a continuous curve), in the extreme case, it may comprise of a single point.

Example. For a discrete function, by convention, we use "n" instead of "x" to represent the independent variable. The y-value may be expressed as a function of "n", we write this as f(n).

Let f(n)= 2n.
Very simply, when n=0,f(0)=0.
When n=1, f(1)=2.
When n=2, f(2)=4 and so forth.
You get the idea?

To restrict this to a single point, in the extreme case, as mentioned above, we can do something like this.

Let f(n)= 2n where 0<n<=1. Here n is a positive integer and we are restricting it to a single point.
In this instance, f(n) consists of the point (n,f(n))=(1,2) only.

So, to answer your question, yes. Even a single point can have a functional form, it can be part of a larger picture (although not very meaningful). But you may not have seen discrete functions in your experience.
I don't think it was a very good question.

A10. Yes, for sure. It only has to possess the property that f(x)=f(-x). This is called an even-function and it is symmetrical with respect to the y-axis.
Similarly, x-axis symmetry requires that the function satisfies the property, f(y)=f(-y).
You can answer this by giving the parabola as an example.
Consider y=f(x)=x^2. Select a test point, say x=1, x=-1.
In both case, f(1)=1, f(-1)=1. This satisfies the f(x)=f(-x) property. So, it is symmetrical w.r.t. the y-axis.