Chapter 1

1.4 Graphs of Equations

Note: The graph of an equation in x & y is the set of all points (x, y) such that x & y satisfy the given equation.

Determine if the points (-1, 3) and (1, 7) lie on the graph of the equation

To determine if (-1, 3) lies on the graph, it must satisfy the equation , We must determine if the equation when evaluated at (-1, 3) is true:

Thus, the point (-1, 3) lies on the graph of the equation

To determine if (1, 7) lies on the graph, it must satisfy the equation

We must determine if the equation when evaluated at (1, 7) is true:

So the point (1, 7) does not lie on the graph of the equation

For the next example, we will use the standard form for the equation of a circle which is given by:

where (h, k) is the center of the circle and r is the radius of the circle.

In many cases, to get an equation of a circle in standard form one must complete the square. Below is review of completing the square:

The expression can be made into a perfect square by adding the
square of one-half the coefficient of x:

When completing the square in an equation, must be added to both sides of the equation.

Find the center & radius of the circle

To find the center and radius, we must first get the equation in standard form. First we will rearrange some terms and complete the square:

We will add the term to each expression on the left and then add the same values on the right hand side.

simplifying the expression, we have:

To see what the center of the circle is we must first put the equation in standard form. Remember the standard form for the equation of a circle is:

with center (h, k) and radius r.

So our above equation in standard form is:

Thus the center of the circle above is (-2, 4) and the radius = .