PRECALCULUS I
Graphs and Lines
•Intercepts, symmetry, circles
•Slope, equations, parallel, perpendicular
Dr. Claude S. Moore
Danville Community College
1
Graph of an Equation
Equation - equality of two quantities.
Solution - (a,b) makes true statement when
a and b are substituted into equation.
Point-plotting method - simplest way to
graph.
x
-2 -1 0 1 2
y = 2x - 3 -7 -5 -3 -1 1
Finding Intercepts
of an Equation
The x-intercept is point where graph touches
(or crosses) the x-axis.
The y-intercept is point where graph touches
(or crosses) the y-axis.
1. To find x-intercepts, let y be zero and
solve the equation for x.
2. To find y-intercepts, let x be zero and
solve the equation for y.
Tests for Symmetry
1. The graph of an equation is symmetric with
respect to the y-axis if replacing x with -x yields
an equivalent equation.
2. The graph of an equation is symmetric with
respect to the x-axis if replacing y with -y yields
an equivalent equation.
3. The graph of an equation is symmetric with
respect to the origin if replacing x with -x and y
with -y yields an equivalent equation.
Standard Form of the
Equation of a Circle
The point (x, y) lies on the circle of
radius r and center (h, k)
if and only if
(x -
2
h)
+ (y -
2
k)
=
2
r
.
Slope-Intercept Form
of the Equation of a Line
The graph of the equation
y = mx + b
is a line whose slope is m and
whose y-intercept is (0, b).
Definition: Slope of a Line
The slope m of the nonvertical line
through (x1, y1) and (x2, y2)
where x1 is not equal to x2 is
y2  y1
m
x2  x1
Point-Slope Form
of the Equation of a Line
The equation of the line with
slope m passing through the
point (x1, y1) is
y - y1 = m(x - x1).
Equations of Lines
1. General form:
2. Vertical line:
3. Horizontal line:
4. Slope-intercept:
5. Point-slope:
1. Ax + By + C = 0
2. x = a
3. y = b
4. y = mx + b
5. y  y1 = m(x  x1)
Parallel and
Perpendicular Lines
Parallel: nonvertical l1 and l2 are
parallel iff m1 = m2 and b1  b2.
*Two vertical lines are parallel.
Perpendicular: l1 and l2 are
perpendicular iff
m1 = -1/m2 or m1 m2 = -1.