Graphs of Equations

When you are done with this section, you will be able to do the following:

Find the x-intercept(s) of an equation (if there are any)
Find the y-intercept(s) of an equation (if there are any)
Determine if an equation is symmetric with respect to the x-axis, symmetric with respect to the y-axis, symmetric with respect to the origin, or has none of these symmetries
Find the standard form of the equation of a circle, the center of a circle, and the radius of a circle
Solve application problems related to x-intercepts and/or y-intercepts

What is an x-intercept?

An x-intercept is a point on the graph of an equation whose y-coordinate is 0. This definition will also let us know how to find any x-intercept of an equation. We just plug in 0 for each of the y variable and then solve for x. Once we have values for x, we put them into an ordered pair of the form (x, 0).

Example 1:

Find the x-intercepts of the graph of the equation

Solution:

To find the x-intercepts, we plug 0 in for all the y variables in the equation.

Now we solve the equation for x. Since this is a quadratic equation (highest exponent is 2), we can try to solve this by factoring or using the quadratic formula. We will show both ways.

This is factoring:

This is using the quadratic formula:

The quadratic formula is

where a, b, and c come from

.
In our problem

, and

. When we plug this into the quadratic formula, we get

Now we have two x values, but we do not yet have the x-intercepts. x-intercepts are points and must be written as ordered pairs.

Thus the x-intercepts are (2, 0) and (-3, 0).

Example 2: This one is for you to try.

Find the x-intercepts of the graph of the equation

.
Solution: Click here for the solution.

It is assumed that you have already been introduced to the x-intercepts of the graphs of equation. This is intended to be a quick review of this topic. There are several sources on-line with additional information and examples related to x-intercepts.

Additional On-line Resources:

What is a y-intercept?

A y-intercept is a point on the graph of an equation whose x-coordinate is 0. This definition will also let us know how to find any y-intercept of an equation. We just plug in 0 for each of the x variable and then solve for y. Once we have values for y, we put them into an ordered pair of the form (0, y).

Example 1:

Find the y-intercepts of the graph of the equation

Solution:

To find the y-intercepts, we plug 0 in for all the x variables in the equation.

Now we solve the equation for y. We get

.
Now we have a y value, but we do not yet have the y-intercept. y-intercepts are points and must be written as ordered pairs.

Thus the y-intercepts is (0, -6).

Example 2: This one is for you to try.

Find the y-intercepts of the graph of the equation

.
Solution: Click here for the solution.

It is assumed that you have already been introduced to the distance formula. This is intended to be a quick review of this formula. There are several sources on-line with additional information and examples related to the distance formula.

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What is symmetry?

Symmetry is an exact correspondence of form and constituent configuration of a graph on the opposite side of a dividing line or about a point.

This is a lovely definition, but what does it mean? Let's look at our specific types of symmetry to answer this question. We are going to study:

symmetry with respect to the x-axis

symmetry with respect to the y-axis

symmetry with respect to the origin (the point (0, 0))

Each of these symmetries tells us the dividing line or point on which we are going to look on the opposite side to determine symmetry.

x-axis symmetry:

A graph is symmetric with respect to the x-axis if when we fold the graph along the x-axis, the part of the graph below the x-axis matches exactly with the part of the graph above the x-axis. This description exaplains how you would look at a graph and determine graphically if it is symmetric with respect to the x-axis.

What does this mean algebraically? Since we are talking about above and below the x-axis, we are talking about y values. Thus leads us to think about comparing what happens to a graph when we put in two different y values (the positive one and the respective negative one). A graph of an equation is symmetric with respect to the x-axis if whenever the point (x, y) is on the graph, the point (x, -y) is also on the graph.

Example 1:

Determine if the graph of the equation is symmetric with respect to the x-axis.

Solution:

To determine if a graph is symmetric with respect to the x-axis, we want to see if when we replace all y with -y in the equation, the equation does not change.

Now we compare this equation with the original equation . You may be able to just look at these two equation and see that they are not the same. This would indicate that the graph of the equation is not symmetric with respect to the x-axis.

If you cannot just look at the equations and see that they are the same, then you can replace y in one of the equations with the x part of the other equation and see if you can simplify the equation to 0 = 0.

We can see that we did not end up at 0 = 0, thus the graph of is not symmetric with respect to the x-axis.

y-axis symmetry:

A graph is symmetric with respect to the y-axis if when we fold the graph along the y-axis, the part of the graph to the right of the y-axis matches exactly with the part of the graph to the left of the y-axis. This description explains how you would look at a graph and determine graphically if it is symmetric with respect to the y-axis.

What does this mean algebraically? Since we are talking about to the left of and to the right of the y-axis, we are talking about x values. Thus leads us to think about comparing what happens to a graph when we put in two different x values (the positive one and the respective negative one). A graph of an equation is symmetric with respect to the y-axis if whenever the point (x, y) is on the graph, the point (-x, y) is also on the graph.

Example 2:

Determine if the graph of the equation is symmetric with respect to the y-axis.

Solution:

To determine if a graph is symmetric with respect to the y-axis, we want to see if when we replace all x with -x in the equation, the equation does not change.

Now we compare this equation with the original equation . You may be able to just look at these two equation and see that they are not the same. This would indicate that the graph of the equation is not symmetric with respect to the y-axis.

If you cannot just look at the equations and see that they are the same, then you can replace y in one of the equations with the x part of the other equation and see if you can simplify the equation to 0 = 0.

We can see that we did not end up at 0 = 0, thus the graph of is not symmetric with respect to the y-axis.

origin symmetry:

A graph is symmetric with respect to the origin is when we rotate the graph 180 degrees, the image of the graph looks the same as the original. This description explains how you would look at a graph and determine graphically if it is symmetric with respect to the origin.

What does this mean algebraically? Since we are talking about rotating 180 degrees, we are talking about x values and y values. Thus leads us to think about comparing what happens to a graph when we put in two different x values (the positive one and the respective negative one) and two different y values (the positive one and the respective negative one). A graph of an equation is symmetric with respect to the origin if whenever the point (x, y) is on the graph, the point (-x, -y) is also on the graph.

Example 3:

Determine if the graph of the equation is symmetric with respect to the origin.

Solution:

To determine if a graph is symmetric with respect to the origin, we want to see if when we replace all x with -x in the equation and all y with -y in the equation, the equation does not change.

Now we compare this equation with the original equation . You may be able to just look at these two equation and see that they are not the same. This would indicate that the graph of the equation is not symmetric with respect to the y-axis.

If you cannot just look at the equations and see that they are the same, then you can replace y in one of the equations with the x part of the other equation and see if you can simplify the equation to 0 = 0.

We can see that we did not end up at 0 = 0, thus the graph of is not symmetric with respect to the origin.

Example 4: This one if for you to try.

Algebraically check for symmetry of the graph of the equation . This means determine if the graph of the equation is symmetric with respect to the x-axis, determine if the graph of the equation is symmetric with respect to the y-axis, and determine if the equation is symmtric with respect to the origin.
Solution: Click here for the solution.

Example 5: This one if for you to try.

Algebraically check for symmetry of the graph of the equation . This means determine if the graph of the equation is symmetric with respect to the x-axis, determine if the graph of the equation is symmetric with respect to the y-axis, and determine if the equation is symmtric with respect to the origin.
Solution: Click here for the solution.

Example 6: This one if for you to try.

Algebraically check for symmetry of the graph of the equation

. This means determine if the graph of the equation is symmetric with respect to the x-axis, determine if the graph of the equation is symmetric with respect to the y-axis, and determine if the equation is symmtric with respect to the origin.

Solution: Click here for the solution.

Example 7: This one if for you to try.

Algebraically check for symmetry of the graph of the equation

Example 8: This one if for you to try.

Graphically check for symmetry of the graph of the equation

Solution: Click here for the solution.

Example 9: This one if for you to try.

Complete the following graph to make it symmetric with respect to the x-axis.

Solution: Click here for the solution.

Example 10: This one if for you to try.

Complete the following graph to make it symmetric with respect to the y-axis.

Solution: Click here for the solution.

Additional On-line Resources:

Circles and what is this standard form thing?

Let's start with what is a circle. We all have a basic idea of what a circle is, but what is the actual definition?
A circle is defined by Merriam-Webster Online as

"a closed plane curve every point of which is equidistant from a fixed point within the curve"

In other words, a circle is a set of points that are the same distance (the radius of the circle) away from a particular point (the center of the circle). If we call the center of the circle (h, k) and the radius r, we can use the distance formula to find the standard form of the equation of a circle.

the distance formula is

Using a generic point (x, y), the center of the circle (h, k), and the radius of the circle r, we get:

This gives us the standard form of an equation of a circle: where (h, k) is the center of the circle and r is the radius of the circle.

Example 1:

Find the standard form of the equation of a circle with center (3, -2) and solution point (-1, 1).

Solution:

In order to write a circle in standard form, we need to center and the radius. The problem gives us the center as (3, -2). All we need now is the radius. There are a couple of ways that we can use the solution point to help with that. Let's do the easiest one first. A solution point means that when I plug the x and y values of the point into my circle equation for x and y, I get a true equation. Thus if I plug in my center and my solution point, I can solve for r.

This gives us

Whenever we take the square root of both sides of an equation, we get both a positive and negative answer. Since r is a radius which is a length, r must be positive. Thus r = 5.

Now we have the center (3, -2) and the radius r = 5. This gives us the equation of the circle in standard form:

The other way to solve this problem is to find the distance between the center and the point on the circle. This would be the radius. We would then plug this along with the center into the standard form of the equation of a circle.

Let's find the distance between the center (3, -2) and the point on the circle (-1, 1).

Thus we know the radius is r = 5. We now plug this into the standard equation with the center to get:

Example 2: This one if for you to try.
Find the standard form of the equation of a circle with a diameter whose endpoints are (-3, 4) and (5, -2).

Solution: Click here for the solution.

Example 3:

Rewrite the equation of the circle in standard form. What is the center of the cirlce? What is the radius of the circle?

Solution:

The process of rewriting the general form of the equation of a circle into standard form requires completing the square. We want to add in exactly the correct numbers so that we can factor the x parts of the equation into a perfect square trinomial that can be factored into and the y parts into a perfect square trinomial that can be factored into . The number that is not part of these perfect squares, will the the square of the radius.

Let's get started. The first step in completing the square is to make the coefficient on the square terms ( and ) to be 1. In this particular problem, we will do that by dividing the equation by 3. This will give us:

We now want to group together the parts with x, group together the parts with y, and move the part that is just a number to the other side of the equation. This will give us:

Now we will complete the square for the x part and for the y part.

To recap in words what we did: You take the coefficient of the x term; divide it by 2; and square that number. You then add this to both sides of the equation. You go through the same process with the coefficient of the y term. We then factor the x part; factor the y part; and simplify the right hand side.

This gives us the standard form of the equation of the circle as:

We are not done yet. We were also asked for the center and the radius. The center can be read from the equation and is (2, -1). The radius is the positive square root of the number on the right hand side of the equation and is .

Note that it is possible to enter in WeBWorK using sqrt(2).

Example 4: This one if for you to try.

Rewrite the equation of the circle in standard form. What is the center of the cirlce? What is the radius of the circle?

Solution: Click here for the solution.

Example 5: This one if for you to try.

Rewrite the equation of the circle in standard form. What is the center of the cirlce? What is the radius of the circle?

Solution: Click here for the solution.

Additional On-line Resources:

Graphs of Equations Homework:

Graphs of Equations Homework contains 7 problems.
Things to remember:

Intercepts are points and must be written as ordered pairs (x-intercepts look like (x, 0) and y-intercepts look like (0, y)).
You should try to do all problems algebraically (without a graphing calculator). This includes algebraically finding intercepts and determining symmetries.
When completing the square, you first need to make the coefficents of and to be 1.
Problem 7 is asking you to find x-intercepts using your graphing calculator. You want to make sure your window is large enough to see all the x-intercepts. Click here for information about how to use your graphing calculator.

Homework examples:

Example 1: similar to problem 2

Keep in mind that it is recommended that you also complete all the problems in the set called "also recommended" with the same number as the Graphs of Equations Homework. There are many different ways to ask the same questions. This will allow you to see additional problems that are related to this topic.