## P.1 Graphs and Models

### The Graph of an Equation

Points are solutions to an equation. The set of all the solutions to an equation is the graph of the equation. Like in algebra, we’ll be working primarily with two-variable equations. The equation $3x+y=7$ has an infinite number of solutions, two of which are $(1,4)$ and $(2,1)$. If you were to plot a number of those solutions, you would see a straight line. The more points, the more accurate your graph will be compared to reality. Just keep in mind that only plotting a few points won’t give you an accurate picture.

### Intercepts of a Graph

If you’re not using a graphing utility, you might get unlucky with the points you choose to plot and end up misrepresenting the equation’s graph. So, you want to focus on finding specific characteristics of graphs. One of those is the intercepts, where the graph crosses the axes. The $x$-intercept is where the graph crosses the $x$-axis and whose point is in the form $(x,0)$. The $y$-intercept is the opposite. It crosses the $y$-axis and has a point in the form $(0,y)$.

You’ve mostly dealt with functions in algebra, and those will only have one $y$-intercept, which is pretty easy to find. The $x$-intercepts, or zeros, are tougher. The equation $y = x^2 + 5x + 6$ is typically factored into the form $y=(x+2)(x+3)$, where you can see it has two zeros when $x = -3$ and $x= -2$.

### Symmetry of a Graph

Another characteristic of graphs is symmetry. Not at all of them have it, but the test for it is fairly quick.

For symmetry with respect to the $y$-axis, plugging in $-x$ will yield the same result as $x$. The equation $y=x^2$ is an example this type of symmetry.

Symmetry with respect to the $x$-axis is the same, save for swapping the variables. These are not functions (one $x$ value is paired with multiple $y$ values), so you don’t see them too often.

Symmetry with respect to the origin has you replace both variables with their negatives. The equation $y=x^3$ is one, since $-y=-x^3$ is equivalent after you factor out the negative.

### Points of Intersection

If you need a solution to multiple equations, you are looking for an intersection point, somewhere where both equations share a solution point. Substitution and elimination are two ways to find these, but using a graphing utility is also valid, as long as it’s available to you.

### Fitting Mathematical Models to Data

I’ll go over how to do this in class, but here’s a step-by-step in case you miss it. This won’t appear on any quizzes or tests, but spreadsheets are a tool worth learning.