Two lines are parallel
if their slopes are equal. Two lines are perpendicular
if their slopes are negative reciprocals of each other. For e.g.
is perpendicular to the line with equation
Ex. In the -plane above, the slope of the line passing through the points and is
Line QR appears to intersect the y-axis close to the point so the -intercept of the line must be close to . To get the exact value of the -intercept, substitute the coordinates of any point on the line, say into the equation , and solve it for as follows.
Therefore, the equation of line QR is .
You can see from the graph that the -intercept of line QR is 2, since QR passes through the point . More generally, you can find the -intercept by setting in an equation of the line and solving it for as follows.
Graphs of linear equations can be used to illustrate solutions of systems of linear equations and inequalities.
Ex. Consider the following system of linear inequalities.
Solving each inequality for in terms of yields
Each point (x, y) that satisfies the first inequality is either on the line or below the line because the y-coordinate is either equal to or less than . Therefore, the graph of consists of the line and the entire region below it. Similarly, the graph of consists of the line and the entire region above it. Thus, the solution set of the system of inequalities consists of all of the points that lie in the shaded region shown in the figure below, which is the intersection of the two regions described.
Symmetry with respect to the x-axis, the y-axis, and the origin is mentioned above.
Another important symmetry is symmetry with respect to the line with equation . The line passes through the origin, has a slope of 1, and makes a 45-degree angle with each axis. For any point with coordinates , the point with interchanged coordinates is the reflection of about the line ; that is, and are symmetric about the line . It follows that interchanging and in the equation of any graph yields another graph that is the reflection of the original graph about the line .
Ex. Consider the line whose equation is . Interchanging and in the equation yields . Solving this equation for yields . The line and its reflection are graphed below.
The line is a line of symmetry for the graphs of and .
The graph of a quadratic equation of the form
are constants and
, is a parabola
. The x
-intercepts of the parabola are the solutions of the equation
is positive, the parabola opens upward and the vertex
is its lowest point. If
is negative, the parabola opens downward and the vertex is the highest point. Every parabola is symmetric with itself about the vertical line that passes through its vertex. In particular, the two
-intercepts are equidistant from this line of symmetry.
Ex. The equation has the following graph.
The graph indicates that the -intercepts of the parabola are –1 and 3. The values of the -intercepts can be confirmed by solving the quadratic equation to get and . The point (1, –4) is the vertex of the parabola, and the line is its line of symmetry. The -intercept is the -coordinate of the point on the parabola at which , which is .
Ex. The graph of is a circle with its center at the origin and with radius 10, as shown in the figure below. The smaller circle has center (6, –5) and radius 3, so its equation is .