Download 3.3-3.4 PPT

3.3 SLOPE AND RATE OF
CHANGE
Geometry R/H
What does each of the following look like?
Positive Slope
Negative Slope
Zero Slope
Undefined Slope
When given 2 points (x1, y1) and (x2, y2)
plug them into our slope formula:
y2  y1
m
x2  x1
Ex1: (4,3)and (2,5)
From a graph!
Find the slope of the line.
Blue Line:
Red Line:
Rate of Change
COLLEGE ADMISSIONS In 2004, 56,878 students applied to UCLA.
In 2006, 60,291 students applied. Find the rate of change in the
number of students applying for admission from 2004 to 2006.
X – independent variable
Y – Dependent variable
Let's try One More
• Find the rate of change for the data in the table.
Pairs of Lines
• Given two lines in the coordinate plane, they
can intersect, coincide, or be parallel.
y = 5x + 8
y = 2x – 5
y = 5x – 4
y = 4x + 3
Parallel lines Intersecting
Same slope
Different
y-intercept
Different
Slopes
y = 2x – 4
y = 2x – 4
Coinciding
Same slope
Same
y-intercept
 We can use this information to compare lines
and then classify them.
Perpendicular Lines
• With intersecting lines, we can also tell if they
are perpendicular or not.
Example:
y = 2x + 4 and
y = -½x - 3
Perpendicular lines
The product of the slopes is -1
The y-intercepts can be any number.
More Parallel and Perpendicular lines
Determine whether AB and CD are parallel,
perpendicular, or neither for the given set of points.
Ex 1: A(1, -3) B(-2, -1) C(5, 0) and D(6, 3)
Ex 2: A(3, 6) B(-9, 2) C(5, 4) and D(2, 3)
Using Slope to Solve Problems
 Determine whether triangle ABC is a right
triangle.
y
Slope of AB is: 2 3
Slope of AC is:  3 2
B
A
0
Since the slopes are
C
the opposite reciprocals
of each other, the line
segments are perpendicular and the triangle is a right triangle.
x
Classifying Pairs of Lines
Determine whether the lines are parallel,
intersect, coincide, or are perpendicular.
2y – 4x = 16,
y – 10 = 2(x - 1)
GRAPHING
Graphing Equations
 Determine which two lines are parallel and
then graph the lines.
y
1
y  x 2
3
y  3x  1
1
y  x 1
3
0
x
Graphing Equations of Lines
Graph this equation:
y
2
y  x2
3
Step 1: Graph the yintercept
Step 2: Use the slope to
find the next point.
Step 3: Draw a line through
the two points.
0
x
Graphing Equations
Graph this line:
2
y  1    x  2
3
The equation is given in
the point-slope form, with
2
3
a slope
of through
the point (–2, 1). Plot the
point (–2, 1) and then rise
–2 and run 3 to find

y
another point. Draw the
line containing the points.
0
x
WRITING EQUATIONS OF
LINES
The Forms of Equations of Lines
• The equation of a line can be written in several different
forms.
• The standard form is Ax + By = C
• The slope-intercept form is y = mx + b, where m is the
slope and b is the
y-intercept.
• The point-slope form is y – y1 = m(x – x1), where m is the
slope and ( x1, y1) is a point on the line.
The Forms of Equations of Lines
• Some special cases are the following:
• The equation of a vertical line is x = a, where a is the x-
intercept. The slope is undefined.
• The equation of a horizontal line is
the y-intercept. The slope is zero.
y = b, where b is
Writing Equations in the SlopeIntercept Form: y = mx +b
1.
Given a slope of 0 and a y-intercept at (0, 5).
2.
Given an undefined slope and passing
through (3, 0).
3.
Given a slope of -4 and through the point
(5, 26).
Given a Slope and a Point
Write the equation of the line with slope 6
through (3, –4) in slope-intercept form
y  y1  m  x  x1 
Start with Point-slope form
y  4  6  x  3 
Substitute 6 for m, 3 for
x 1, and -4 for y 1.
y  4  6 x  18
y  6 x  22
Simplify
Convert to slope-intercept
form.
Write equation in point-slope form.
• (4, 7) and (6, 13)
• (2, -1) and (5, -13)
• (-8, 5) and (-5, 5)
Writing equations of parallel/Perpendicular
Lines
• Original Line (-6, 12) and (3, 6)
• Parallel through (12, -12)
• Perpendicular through (6, 14)
Parallel/Perpendicular Ex 2
• Original line: (5, 27) and (9, 47)
• Parallel through (8, 36)
• Perpendicular through (-20, 11)
MIDPOINT AND EQUATION
OF A PERPENDICULAR
BISECTOR
Review of Midpoint Formula
• You can find the midpoint of a segment by using
the coordinates of its endpoints.
• The midpoint of the segment joining the points
A(x1, y1) and B(x2, y2) has these coordinates:
 x1  x2 y1  y2 
,


2 
 2
Example: Find the midpoint of A (-1, 4) & B (3, 5).
1
2 9

 1  3 4  5 
,

   2 , 2    1, 4 2 
2 




 2
Example 2
• S is the midpoint of RT .
R has coordinates
(-6, -1), and S has coordinates (-1, 1). Find the
coordinates of T.
Step 1: Let the coordinates of T equal (x, y).
Step 2: Use the Midpoint Formula:
 1, 1
 6  x 1  y 
 
,

2 
 2
T   4, 3
Step 3: Find the x- and y-coordinate:
6  x
1 
x4
2
1  y
1
 y 3
2
Find equation of the perpendicular
bisector
• Write the equation for the perpendicular
bisector of segment AB .
• Step 1: Find the midpoint of AB.
04 53 4 8
 2 , 2    2,2

 

  2,4 
6
A
4
B
2
5
 Step 2: Find the slope of AB .
y 2  y1 3  5
2
1

 
x 2  x1 4  0
4
2
Find equation of the perpendicular
bisector
• Step 3: Find the slope of
perpendicular bisector.
• The slope of AB is -½, so the
slope of ⊥ bisector is 2.
(opposite reciprocal)
• Step 4: Use (2, 4) and slope of 2
to find equation of line.
6
A
4
B
2
 Use the point slope form:
y  4  2(x  2)
y  y1  m(x  x1 )
y  2x
5
Now You Try!
Write an equation in point-slope form for the
perpendicular bisector of the segment with
endpoints A(7, 9) and B(–3, 5) .
Step 1: Find the midpoint of AB
Step 2: Find the slope of AB
Step 3: Find the slope of the perpendicular
line.
Step 4: Find the equation of the
perpendicular bisector.
Median of a Triangle
• A median of a triangle is a
segment whose end-points are a
vertex of the triangle and the
mid-point of the opposite side.
C
• What would you need to find an
equation of a median?
A
D
• Every triangle has three medians – one
from each vertex.
B
Altitude of a Triangle
• An altitude of a triangle is a perpendicular segment from
a vertex to the line containing the opposite side.
• Every triangle has three altitudes – one from
each vertex
• An altitude can be inside, outside, or on the
triangle.
A
B
D
C
Examples of Altitudes
• These examples of altitudes show how the altitude can
be inside the triangles, outside the triangle or on (one
of the sides) of a triangle.
• What would you need to find an equation of an
altitude?
a
a
a