Transcript Slide 1

Submitted to - Sh.Bharat
Bhushan Sir
Given the Slope and Y-Intercept
• Just plug them into y = mx + b
• Example: Write the equation for the line with
slope m = 3/2 and y-intercept b = -1
Given a Graph
• Find the slope and y-intercept.
• Then, plug into y = mx + b
• Example:
• Write an equation of
the line shown.
Point-Slope Form
• y – y1 = m(x – x1)
• Used when you are given:
▫ The slope and a point
▫ Two points
▫ A parallel or perpendicular line and a point
Parallel and Perpendicular Lines
• For parallel lines:
– use the same slope
• For perpendicular lines:
– use the opposite reciprocal (flip it and
change the sign)
• Then use Point-Slope form
Linear Equations
1) linear equation
2) linear function
3) standard form
4) y-intercept
5) x-intercept
Linear Equations
y
6
Sara has 4 hours after dinner to study and do
homework. She has brought home algebra 2
and chemistry.
4
x y 4
2
If she spends x hours on algebra and y
hours on chemistry, a portion of the graph
of the equation x + y = 4 can be used to
relate how much time she spends on each.
x
0
-2
-2
2
0
An equation such as x + y = 4 is called a linear equation.
4
6
Linear Equations
A linear function is a function whose ordered pairs satisfy a linear equation.
Any linear function can be written in the form
are real numbers.
f(x) = mx + b,
where m and b
State whether each function is a linear function. Explain.
a)
f(x) = 10 – 5x
Yes!
It can be written as f(x)
m = – 5, b = 10
b)
g(x) = x4 – 5
No!
x
c)
h(x, y) = 2xy
= – 5x + 10
has an exponent other than 1.
No!
Two variables are multiplied together.
Linear Equations
Standard Form of a Linear Function
The standard form of a linear function is
Ax + By = C,
where A > 0,
A and B are not both zero.
Also, A, B, and C are integers whose greatest common factor is 1
Write the equation in Standard Form:
y  3x  4
3x  y  4
Linear Equations
In the previous lesson, you graphed functions by using a table of values.
Since two points determine a line, there are quicker ways to graph linear functions.
One way is to find the points at which the graph intersects each axis and then connect
them with a line.
y
The y-coordinate of the point at which the graph
y - intercept
crosses the y-axis is called the ____________.
(0, 3)
(-4, 0)
x
The x-coordinate of the point at which the graph
x - intercept
crosses the x-axis is called the ____________.
Linear Equations
Find the x-intercept and the y-intercept of the graph of the equation.
Then graph the equation.
5x  3 y  15
The x-intercept is the value of x when y = 0.
5x  3(0)  15
y
(0, 5)
5 x  15
x3
The y-intercept is the value of y when x = 0.
5(0)  3 y  15
3 y  15
y 5
x
(3, 0)
FORMULA FOR FINDING SLOPE
The formula is used when you know two
points of a line.
They look like A( X1 , Y1 ) and B( X 2 , Y2 )
RISE X 2  X 1 X 1  X 2
SLOPE 


RUN Y2  Y1
Y1  Y2
Find the slope of the line between the two points (-4, 8) and (10, -4)
If it helps label the points.
Then use the
formula
X 1 Y1
X2
Y2
X 2  X1
(10)  (4)
Y2  Y1 SUBSTITUTE INTO FORMULA (4)  (8)
(10)  (4)
10  4
14
7
Then Sim plify



(4)  (8)  4  (8)  12
6
Graphing Linear Equations
In Slope-Intercept Form
We have already seen that linear
equations have two variables and when
we plot all the (x,y) pairs that make the
equation true we get a line.
In this section, instead of making a
table, evaluating y for each x, plotting
the points and making a line, we will
use The Slope-Intercept Form of the
equation to graph the line.
These equations are all in SlopeIntercept Form:
y  2x  1
y  x  4
3
y x2
2
Notice that these equations are all
solved for y.
Just by looking at an equation in this form, we
can draw the line (no tables).
•The constant is the y-intercept.
•The coefficient is the slope.
y  2x  1
Constant = 1, y-intercept = 1.
y  x  4
Constant = -4, y-intercept = -4.
3
y x2
2
Coefficient = 2, slope = 2.
Coefficient = -1, slope = -1.
Constant = -2, y-intercept = -2.
Coefficient = 3/2, slope = 3/2.
The formula for Slope-Intercept Form is:
y  mx  b;
• ‘b’ is the y-intercept.
• ‘m’ is the slope.
On the next three slides we will graph the three
equations:
3
y  2x 1, y  x  4, y  x  2
2
using their y-intercepts and slopes.
y  2x  1
right 1
1) Plot the y-intercept as a point on
the y-axis. The constant, b = 1, so
the y-intercept = 1.
right 1 up 2
up 2
2) Plot more points by counting the
slope up the numerator (down if
negative) and right the
denominator. The coefficient, m =
2, so the slope = 2/1.
Important!!!
This is one of the big concepts in
Algebra 1. You need to thoroughly
understand this!
Slope – Intercept Form
y = mx + b
m represents the slope
b represents the y-intercept
Writing Equations
When asked to write an equation, you
need to know two things – slope (m)
and y-intercept (b).
There are three types of problems you
will face.
Writing Equations – Type #1
Write an equation in slope-intercept form of the
line that has a slope of 2 and a y-intercept of
6.
To write an equation, you need two things:
slope (m) = 2
y – intercept (b) = 6
We have both!! Plug them into slope-intercept
form
y = mx + b
y = 2x + 6
Writing Equations – Type #2
Write an equation of the line that has a slope of
3 and goes through the point (2,1).
To write an equation, you need two things:
slope (m) = 3
y – intercept (b) = ???
We have to find the y-intercept!! Plug in the
slope and ordered pair into
y = mx + b
1 = 3(2) + b
Writing Equations – Type #2
1 = 3(2) + b
Solve the equation for b
1=6+b
-6 -6
-5 = b
To write an equation, you need two things:
slope (m) = 3
y – intercept (b) = -5
y = 3x - 5
Writing Equations – Type #3
Write an equation of the line that goes through the
points (-2, 1) and (4, 2).
To write an equation, you need two things:
slope (m) = ???
y – intercept (b) = ???
We need both!! First, we have to find the slope. Plug
the points into the slope formula.
Simplify
2 1
m
4  (2)
1
m
6
Writing Equations – Type #3
Write an equation of the line that goes through the
points (-2, 1) and (4, 2).
To write an equation, you need two things:
1
slope (m) = 6
y – intercept (b) = ???
It’s now a Type #2 problem. Pick one of the ordered
pairs to plug into the equation. Which one looks
easiest to use?
I’m using (4, 2) because both numbers are positive.
1
2 = (4) + b
6
Writing Equations – Type #3
1
2 = (4) + b
6
Solve the equation for b
2
2= +b
3
2
2
 
3
3
1
1 b
3
To write an equation, you need two things:
1
slope (m) =
6
1
y
y – intercept (b) = 1
3
1
1
x 1
6
3