Transcript IB Mathematical Studies - peacock

Review
Linear Equations and Graphs
Linear Equations in Two
Variables
• A linear equation in two variables is an equation that can be
written in the standard form Ax + By = C,
– where A, B, and C are constants (A and B not both 0), and
– x and y are variables.
– Linear equations graph lines. Anything not straight is called a
curve.
• A solution of an equation in two variables is an ordered pair of real
numbers that satisfy the equation.
– For example, (4,3) is a solution of 3x - 2y = 6.
• The solution set of an equation in two variables is the set of all
solutions of the equation.
• The graph of an equation is the graph of its solution set.
Linear Equations
1.
A linear equation can be written in the forms:
STANDARD FORM:
𝐴𝑥 + 𝐵𝑦 = 𝐶
where A, B, and C are integers and x and y are variables.
(whole numbers)
SLOPE-INTERCEPT FORM:
y  mx  b
where m is the slope and b is the y-intercept.
POINT-SLOPE FORM:
y  y1  m( x  x1 )
where m is the slope and (x1,y1) is a point.
2.
A linear equation graphs a straight line.
Graphing Linear Equations
What is Intercept in Math?
Y
X
An Axis Intercept is the point
where a line crosses the x or
y axis
Intercepts of a Line
x-intercept: where the graph crosses the
x-axis. The coordinates are (a, 0).
To find the x-intercept, let y = 0 and solve for x.
y-intercept: where the graph crosses the
y-axis. The coordinates are (0, b).
To find the y-intercept, let x = 0 and solve for y.
Introduction Linear Graphs
• When to use the x and y intercepts to
graph linear equation
Given the equation in Standard Form
Ax + By = C
Introduction Linear Graphs
Finding Intercepts:
In the equation of a line, let y = 0 to find the “x-intercept” and let
x = 0 to find the “y-intercept”.
Note: A linear equation with both x and y variables will have both
x- and y-intercepts.
• Example: Find the intercepts and draw the graph of 2x –y = 4
x-intercept: Let y = 0 : 2x –0 = 4
2x = 4
x=2
y-intercept: Let x = 0 : 2(0) – y = 4 -y = 4
y = -4
x-intercept is (2,0)
y-intercept is (0,-4)
Introduction Linear Graphs
Finding Intercepts:
In the equation of a line, let y = 0 to find the “x-intercept” and let
x = 0 to find the “y-intercept”.
Example: Find the intercepts and draw the graph of 4x –y = -3
x-intercept: Let y = 0 : 4x –0 = -3
4x = -3
x = -3/4
y-intercept: Let x = 0 : 4(0) – y = -3 -y = -3
y=3
3
x-intercept is (- , 0)
4
y-intercept is (0,3)
Example:
2x + 3y = 5
x-intercept
0, 53 
2x + 3(0) = 5
2x + 0 = 5
2x = 5
x
5
2
y-intercept
2(0) + 3y = 5
0 + 3y = 5
3y = 5
y
5
3
 52 , 0
Example:
4x - y = 6
x-intercept
4x - (0) = 6
4x - 0 = 6
4x = 6
x
 32 , 0
3
2
y-intercept
4(0) - y = 6
0-y=6
-y = 6
y  6
0,  6
Example:
4x + 2y = 3
x-intercept
0, 32 
4x + 2(0) = 3
4x + 0 = 3
4x = 3
x
3
4
y-intercept
4(0) + 2y = 3
0 + 2y = 3
2y = 3
y
Jeff Bivin -- LZHS
3
2
 52 , 0
Example:
3x - 2y = 7
x-intercept
3x - 2(0) = 7
3x - 0 = 7
3x = 7
x
73 , 0
7
3
y-intercept
3(0) - 2y = 7
0 - 2y = 7
-2y = 7
y
Jeff Bivin -- LZHS
7
2
0, 27 
Introduction Linear Graphs
Graphing a line that passes through the
origin:
Some lines have both the x- and y-intercepts at the origin.
Note: An equation of the form Ax + By = 0 will always pass
through the origin. Find a multiple of the coefficients of x and y and
use that value to find a second ordered pair that satisfies the
equation.
• Example:
A) Graph x + 2y = 0
The Slope of a Line
Finding the slope of a line given an equation of
the line:
The slope can be found by solving the equation such
that y is solved for on the left side of the equal sign.
This is called the slope-intercept form of a line. The
slope is the coefficient of x and the other term is the yintercept. The slope-intercept form is
y = mx + b
• Example : Find the slope of the line given 3x – 4y = 12
3 x  4 y  12
4 y  3 x  12
3
y  x3
4
3
 The slope is
4
The Slope of a Line
Finding the slope of a line given an equation of
the line:
• Example : Find the slope of the line given y + 3 = 0
y = 0x - 3
The slope is 0
• Example : Find the slope of the line given x + 6 = 0
Since it is not possible to solve for y, the slope is “Undefined”
Note: Being undefined should not be described as “no slope”
• Example : Find the slope of the line given 3x + 4y = 9
3x  4 y  9
3
4 y  3x  9  The slope is - 4
3
9
y  x
4
4
Introduction Linear Graphs
Recognizing equations of vertical and
horizontal lines:
An equation with only the variable x will always intersect the x-axis
and thus will be vertical.
An equation with only the variable y will always intersect the y-axis
and thus will be horizontal.
• Example: A) Draw the graph of y = 3
B) Draw the graph of x + 2 = 0
x = -2
A)
B)
Graphing Horizontal Lines
This line has a y value of 4 for any x-value. It’s
equation is
y=4
equals 4)
(meaning y always
Graphing Vertical Lines
This line has a x value of 1 for any y-value. It’s
equation is
x=1
equals 1)
(meaning x always
The Equation of a Vertical Line is
X=Constant
x=1
The Equation of a Horizontal Line
is Y=Constant
y=3
Graph the following lines
Y = -4
Y=2
X=5
X = -5
X=0
Y=0
Answers
x = -5
x=5
Answers
y=2
y = -4
Answers
Horizontal and Vertical lines are
always perpendicular to each other
y=0
x=0
Slope of a Line
SLOPE =
RISE
RUN

Slope is a measure of STEEPNESS
The Symbol for
SLOPE = m
Think of m for Mountain
Slope of a Line
Slope of a line:
 x1, y1 
y2  y1 rise
m

x2  x1 run
rise
run
 x2 , y2 
Note: The slope of a line is the SAME everywhere on the line!!! You may use
any two points on the line to find the slope.
The Slope of a Line
Finding the slope of a line given two points on
the line:
The slope of the line through two distinct points (x1, y1) and (x2, y2) is:
rise change in y y2  y1
slope  m 


run change in x x2  x1
( x 2  x1 )
Note: Be careful to subtract the y-values and the x-values in the same order.
Correct
Incorrect
y2  y1
y1  y2
or
x2  x1
x1  x2
y2  y1
y1  y2
or
x1  x2
x2  x1
The Slope of a Line
Finding the slope of a line given two points on
the line:
• Example : Find the slope of the line through the points (2,-1) and (-5,3)
rise y2  y1 3  (1)
4
4
slope  m 

=


run x2  x1 (5)  2 7
7
m=SLOPE =
RISE
y 2  y1

RUN
x 2  x1
x2y2
(6,4)
4
•

3
x1y1
2
(3,2)
RISE  2
•
RUN  3
1
(0,0)
1
2
3
 4
5
6
y 2  y1 4  2 2
Slope  m 


x 2  x1 6  3 3

Switch points and calculate slope
Make (3,2) (x2,y2) & (6,4) (x1,y1)
(x1,y1)(3,2)
•
(x2,y2)(6,4)
(x1,y1)(6,4)
•
•
(x2,y2)(3,2)
•
Recalculation with points
switched
(x1,y1)(6,4)
•
(x2,y2)(3,2)
•
y 2  y1 4  6 2 2
Slope  m 



x 2  x1 2  5 3 3
Same slope as before
It doesn’t matter what 2 points
you choose on a line
the slope must come out the same
Keeping Track of Signs When Finding
The Slope Between 2 Points
•
Be Neat & Careful
• Use (PARENTHASES)
• Double Check Your Work as you Go
• Follow 3 Steps
3 Steps for finding the Slope of a
line between 2 Points (3,4)&(-2,6)
1st Step: Write x1,y1,x2,y2
over numbers
x1 y1
x2 y2
(3,4) & (-2,6)
2nd Step: Write Formula
and Substitute x1,x2,y1,y2
values.
Slope 
3rd Step: Calculate &

Simplify

y 2  y1 6  4

x 2  x1 2  3
6  4 2
2


2  3 5
5
Find the Slopes of Lines containing
these 2 Points
1. (1,7) & (5,2)
3. (-3,-1) & (-5,-9)
5. (3,6) & (5,-5)
2. (3,5) & (-2,-8)
4. (4,-2) & (-5,4)
6. (1,-4) & (5,9)
ANSWERS
1. (1,7) & (5,2)
y 2  y1 2  7 5
Slope 


x 2  x1 5  1 4
2. (3,5) & (-2,-8)
y 2  y1 8  5 13 13
Slope 



x 2  x1 2  3 5
5
3. (-3,-1) & (-5,-9)
4. (4,-2) & (-5,4)
5. (3,6) & (5,-5)
6. (1,-4) & (5,9)

y 2  y1 9  (1) 8 4
y 2  y1 4  (2) 6
2
Slope 



x 2  x1 5  (3) 2 1 Slope  x  x  5  4  9   3
2
1
y 2  y1 5  6  11
Slope 


x 2  x1
5 3
2
y 2  y1 9  (4) 13
Slope 


x 2  x1
5 1
4
Solve for y if (9,y) & (-6,3) & m=2/3
y 2  y1
Slope 
x 2  x1
2 3  y1 3  y


3 6  9 15


2 3  y1 3  y
(15) 

(15)
3 6  9 15
(5)2  3  y
10  3  y

13   y




13  y
Review Finding the Slopes
of Lines Given 2 Points
1st Step: Write x1,x2,y1,y2 over numbers
2nd Step: Write Formula and Substitute x1,x2,y1,y2 values.
3rd Step: Calculate & Simplify
NOTE:
y 2  y1
m  Slope 
x 2  x1
Be Neat, Careful, and Precise and Check your work as you
go..

RISE
SLOPE  m 
RUN

Positive Slope
Is Up the Hill
Negative Slope
Is Down the Hill
NO Slope
Vertical Drop
ZERO Slope Horizontal
RISE
SLOPE  m 
RUN

NO Slope
Vertical Drop
ZERO Slope Horizontal
RISE any _ number

 Undefined(NO_ Slope)
RUN
0
RISE
0

0
RUN any _ number

Interpreting Slope
Lines that increase from left to right have a Positive slope.
y
x
Lines that decrease from left to right have a Negative slope.
y
x
Interpreting Slope
Lines that are horizontal have a slope of Zero.
y
x
Lines that are vertical have an Undefined slope.
y
x
The Slope of a Line
Graph a line given its slope and a point on the
line:
Locate the first point, then use the slope to find a
second point.
Note: Graphing a line requires a minimum of two points. From the first
point, move a positive or negative change in y as indicated by the value of
the slope, then move a positive value of x.
• Example : Graph the line given
2
slope =
passing through (-1,4)
3
Note: change in y is +2
The Slope of a Line
Graph a line given its slope and a point on
the line:
Locate the first point, then use the slope to find a second point.
• Example : Graph the line given
slope = -4 passing through (3,1)
Note:
A positive slope indicates the line moves up from L to R
A negative slope indicates the line moves down from L to R
The Slope of a Line
Using slope to determine whether two lines are
parallel, perpendicular, or neither:
Two non-vertical lines having the same slope are
parallel.
Two non-vertical lines whose slopes are negative
reciprocals are perpendicular.
• Example : Is the line through (-1,2) and (3,5) parallel to the line through
(4,7) and (8,10)?
For line 1:
52
3
m1 

3  (1) 4
 YES
For line 2:
10  7 3
m2 

84 4
The Slope of a Line
Using slope to determine whether two lines are parallel,
perpendicular, or neither:
Two non-vertical lines having the same slope are parallel.
Two non-vertical lines whose slopes are negative reciprocals are perpendicular.
• Example : Are the lines 3x + 5y = 6 and 5x - 3y = 2 parallel,
perpendicular, or neither? For line 1:
For line 2:
3x  5 y  6
5 y  3x  6
5x - 3 y  2
 3 y  5 x  2
3
6
5
2
y  x
y  x
5
5
3
3
3
5
 is the negative reciprocal of
5
3
 Perpendicular
Linear Equations in Two Variables
• Writing an equation of a line given its slope and yintercept.
• The slope can be found by solving the equation such
that y is solved for on the left side of the equal sign.
• This is called the slope-intercept form of a line. The
slope is the coefficient of x and the other term is the yintercept.
• The slope-intercept form is y = mx + b, where m is the
slope and b is the y-intercept.
• Example: Find an equation of the line with slope 2 and
y-intercept (0,-3)
• Since m = 2 and b = -3, y = 2x - 3
Equations of a Line
There are
3 Forms of Line Equations
Ax+By=C
•
Standard Form:
•
Slope Intercept Form:
•
Point-Slope Form
y=mx+b
y-y1=m(x-x1)
All 3 describe the line completely
but are used for different purposes.
You can convert from one form to
another.
Converting from
Standard Form: ax+by=c
to Slope Intercept Form
3x  6y  12
6y  3x  12



6
3
12
y
x
6
6
6
1
y  x 2
2
JUST
SOLVE
FOR Y
Slope Intercept
Form: y=mx+b
Converting from
Slope Intercept Form: y=mx+b
to Standard Form
Make the
equation
equal to 0
Standard Form:
Ax+By+C=0
Graphing linear Equations
Linear Equations in Two Variables
Graphing a line using its slope and y-intercept:
• Example: Graph the line using the slope and y-intercept: y = 3x - 6
Since b = -6, one point on the line is (0,-6).
Locate the point and use the slope (m
(0+1,-6+3)=(1,-3)
3
=1
) to locate a second point.
y = 2x + 1
y-intercept
b=1
slope
m=2
rise
2

run
1
Run
1
Rise
2
y = -3x + 2
y-intercept
b=2
slope
m = -3
rise
3

run
1
Run
1
Rise
-3
2
y  x 1
3
y-intercept
b = -1
slope
2
m
3
rise
2

run
3
Run
3
Rise
2
1
y
x3
2
y-intercept
b=3
slope
1
m
2
rise
1

run
2
Run
2
Rise
-1
Review Steps of Graphing
from the Slope Intercept
Equation
1.
Make sure equation is in y=mx+b form
2.
Plot b(y-intercept) on graph (0,b)
3.
From b, Rise and Run according to the slope to plot 2nd
point.
4.
Check sign of slope visually
Point-Slope Form
The point-slope form of the equation of a line is
y  y1  m( x  x1 )
where m is the slope and (x1, y1) is a given point.
It is derived from the definition of the slope of a line:
y2  y1
m
x2  x1
Cross-multiply and substitute
the more general x for x2
Linear Equations in Two Variables
Writing an equation of a line given its slope and a point
on the line: The “Point-Slope” form of the equation of a line
with slope m and passing through the point (x1,y1) is:
y - y1 = m(x - x1)
Where m is the given slope and x1 and y1 are the respective values of the given point.
2
• Example: Find an equation of a line with slope
and a given
5
point (3,-4)
y  y1  m( x  x1 )
2
( x  3)
5
2
6
y4 x
5
5
2
6 20 2
26
y  x 
 x
5
5 5 5
5
5 y  2 x  26 or in standard form 2 x  5 y  26
y  (4) 
Find the Equation of a Line (Given Pt. &
Slope) Using the Pt.-Slope Eq.
Given a point (2,5) & m=5 Write the Equation
y  y1  m(x  x1)
y  5  5(x  2)
y  5  5x  10
y  5x  5
y  5x  5
1.
Write Pt.-Slope Equation
2.
2. Plug-in (x,y) & m values
3.
Solve for y
Linear Equations in Two Variables
Writing an equation of a line given two points on the
line: The standard form for a line was defined as Ax+By=C.
• Example: Find an equation of a line passing through the points
(-2,6) and (1,4). Write the answer in standard form.
Step 1: Find the slope:
46
2
2
m


Step 2: Use the point-slope
1  ( 2)
3
3
y  y1  m( x  x1 )
method:
2
y  6   ( x  (2))
3
2
4
y6   x
3
3
2
4 18
2
14
y  x 
 x
3
3 3
3
3
3 y  2 x  14 or 2 x  3 y  14
Parallel Lines
Have the Same Slope
5
RISE  2
4
3
2
1
•
RISE  2
RUN  3

•
RUN  3

(0,0)
1

2
3
4
5
6
Perpendicular Lines
Have Neg. Reciprocal Slopes
3
3
m2  
2
2
m1 
3
2

1
(0,0)
1

2
3
4
5
2
3
m1  m2     1
3
2
6
Linear Equations in Two Variables
Finding equations of Parallel or Perpendicular lines:
If parallel lines are required, the slopes are identical.
If perpendicular lines are required, use slopes that are negative
reciprocals of each other.
• Example: Find an equation of a line passing through the point
(-8,3) and parallel to 2x - 3y = 10.
Step 1: Find the slope
of the given line
Step 2: Use the point-slope method
y  y1  m( x  x1 )
2 x  3 y  10
3 y  2 x  10
y
2
10
x
3
3
 m
2
( x  (8))
3
2
16
y 3 x
3
3
2
16 9 2
25
y  x   x
3
3 3 3
3
3 y  2 x  25 or -2 x  3 y  25
y 3 
2
3
Linear Equations in Two Variables
Finding equations of Parallel or Perpendicular
lines:
• Example: Find an equation of a line passing through the
point (-8,3) and perpendicular to 2x - 3y = 10.
Step 1: Find the slope
of the given line
Step 3: Use the point-slope method
y  y1  m( x  x1 )
2 x  3 y  10
3 y  2 x  10
y
2
10
x
3
3
 m
2
3
Step 2: Take the negative reciprocal
of the slope found
2
3
m
 m1  
3
2
3
y  3   ( x  (8))
2
3
y  3   x  12
2
3
y   x9
2
2 y  3 x  18 or 3x  2 y  18
Summary of Linear Graphs
Forms of Linear Equations
Equation
Description
When to Use
Y = mx + b
Slope-Intercept Form
slope is m
y-intercept is (0,b)
Given an equation, the slope
and y-intercept can be easily
identified and used to graph
y - y1 = m(x-x 1)
Point-Slope Form
slope is m
line passes through (x 1,y1)
This form is ideal to use when
given the slope of a line and
one point on the line or given
two points on the line.
Standard Form
(A,B, and C are integers, A>0)
Slope is -(A/B)
x-intercept is (C/A,0)
y-intercept is (0,C/B)
Horizontal line
slope is 0
y-intercept is (0,b)
X- and y-intercepts can be found
quickly
Ax+By+C=0
y=b
x=a
Vertical line
slope is undefined
x-intercept is (a,0)
Graph intersects only the y
axis, is parallel to the x-axis
Graph intersects only the x
axis, is parallel to the y-axis