Graphs of equations and Symmetry

Introduction

In this section we will discuss the graphs of equations involving two variables in addition to some important features of graphs. These features include the x-intercept, y-intercept, and symmetry of the graph.

Points on a graph

The geometric interpretation of an equation with two variables can be thought of the collection of points (x, y) that satisfy the equation. This allows us to graph or "draw" the points that satisfy the equation. One of the first methods we learn about to graph an equation is to create an x-y table of points that satisfy the equation, and must be on the graph of the equation. It also gives us a way of determining if a point is on the graph, which is equivalent to asking if those values of x and y satisfy the equation. In biology you might have a collection of data points and try to find an equation whose graph goes through those points in some way that is compatible with the trends in the data. What we are doing in this section is that process in reverse.

Example: Determine if (3, 5) is on the graph of the equation 3x + y = 7.

If (3, 5) is on the graph of 3x + y = 7, then replacing x with 3 and y with 5 yields a true statement. To check this we can look at 3(3) + 5 = 7. This means 14 = 7, which is absurd, and thus (3, 5) is not on the graph of 3x + y = 7.

Intercepts

We are always interested in when the graph of an equation intersects the x-axis or y-axis. By a point being in the intersection of the graph of two equations, we mean that it is a point which has coordinates that satisfies both equations. These points are easier to find than others, and, in the case of polynomials, the points where the graph of the equation intersects the x-axis corresponds to zeroes of the polynomial. The points where the graph of the equation intersects the x-axis(y-axis) are(is) called x-intercepts(y-intercept). To find the x-intercepts, we set y to zero, and solve for x. For the y-intercepts, we do the same, setting x to zero and solving for y. If we cannot set x to zero, then there are no y-intercepts. It is very important to remember that intercepts are points.

Example: Find the x and y-intercepts of y = (x - 4)(x + 3).

For the x-intercepts, we set y to zero, so 0 = (x - 4)(x + 3). This means x = 4 or -3. So the x-intercepts are (-3, 0) and (4, 0). For the y-intercept, we set x to zero to find y = (-4)(3) = -12. Thus, the y-intercept is (0, 12).

Symmetry

The symmetry of the graph of an equation is useful when we want to study some of the properties of a graph. If the equation is a model of something, then we can obtain more information by looking at the symmetries of its defining equations.

The two types of symmetry that we are most worried about are symmetry across the y-axis, and symmetry with respect to the origin.

 The symmetry with respect to the y-axis can be described as if you fold the graph across the y-axis will it land on top of itself.
 The algebraic way to describe this is that if (x, y) is on the graph, then so is (-x, y).

 Symmetry with respect to the origin is harder to describe geometrically. The two most common ways to describe it are as a double 
 reflection and a rotation. The double reflection explanation is if you reflect the graph across the y-axis then the x-axis does
 it land on top of itself. The rotation explanation is if you rotate the graph 180 degrees or $\pi$ radians does the
 graph land on top of itself. Algebraically, the graph of an equation is symmetric with respect to the origin if the following 
 holds: if (x, y) is on the graph of the equation, then (-x, -y) is also on the graph.

Example:

What types of symmetry does $y={\frac {4x^{2}}{3x^{2}-1}}$ have?

Solution: For symmetry across the y-axis, we replace x with -x to get ${\frac {4(-x)^{2}}{3(-x)^{2}-1}}={\frac {4x^{2}}{3x^{2}-1}}$. Thus, the equation is symmetric across the y-axis. For checking symmetry across the origin, we see if $-y={\frac {4x^{2}}{3x^{2}-1}}$ is equivalent to the original equation. Since it is not, the equation is not symmetric with respect to the origin.

Return to Topics Page