Transcript alg1sec5-3pt1.pptx

Slope-Intercept Form
Section 5-3 Part 1
Goals
Goal
• To write linear equations
using slope-intercept form.
Rubric
Level 1 – Know the goals.
Level 2 – Fully understand the
goals.
Level 3 – Use the goals to
solve simple problems.
Level 4 – Use the goals to
solve more advanced problems.
Level 5 – Adapts and applies
the goals to different and more
complex problems.
Vocabulary
•
•
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Parent Function
Linear Parent Function
Linear Equation
y-intercept
Slope-Intercept Form
Definition
• Family of Functions - is a set of
functions whose graphs have basic
characteristics in common.
– Example: All linear functions form a
family because all of their graphs are
the same basic shape, a line.
• Parent Function – is the simplest or
basic function within a family of
functions.
• Linear Parent Function – is y = x
or f(x) = x.
Definition
• Linear Equation - is an equation
that models a linear function.
– In a linear equation the variables are
raised to the 1st power only.
– Example:
• Linear equation; y = 2x + 3.
• Not a linear equation; y = x2 or y = 2x.
– The graph of a linear equation contains
all the ordered pairs that are solutions
of the equation.
• y-intercept – is the y-coordinate of
the point where the line crosses the
y-axis.
Slope-Intercept Form of a
Linear Equation
If you know the slope of a line and the y-intercept, you can write
an equation that describes the line.
Step 1 If a line has a slope of m and the y-intercept is b, then (0, b)
is on the line. Substitute these values into the slope formula.
Slope-Intercept Form of a
Linear Equation
Step 2 Solve for y:
Simplify the denominator.
Multiply both sides by x.
mx = y – b
+b
+b
mx + b = y,
Add b to both sides.
or y = mx + b
Slope-Intercept Form of a
Linear Equation
y = mx + b
slope
y-intercept
Any linear equation can be written in slope-intercept form by
solving for y and simplifying. In this form, you can immediately
see the slope and y-intercept. Also, you can quickly graph a line
when the equation is written in slope-intercept form.
Finding the Slope and y-intercept
of a Line
Procedure for finding the slope and y-intercept of a
line.
1) Rewrite the equation of the line in slope-intercept
form by solving for y.
2) The coefficient of x, is the slope and the constant
term, is the y-intercept.
y = mx + b
Slope (always comes
in front of x!)
y – intercept (where the
line crosses the y-axis)
Example: Finding the Slope and
the y-Intercept
Find the slope and the y-intercept of the line whose equation is
2x – 3y + 6 = 0.
Procedure We need to solve for y.
2x – 3y + 6 = 0 This is the given equation.
2x + 6 = 3y
To isolate the y-term, add 3 y on both sides.
3y = 2x + 6
Reverse the two sides. (This step is optional.)
2
Divide both sides by 3.
y  x2
3
The coefficient of x, 2/3, is the slope and the constant term, 2, is
the y-intercept.
Example: Find the slope and
y-intercept of the equation.
(c) x = -2
(a) y = ½ x + 5
(b) y = 3
y = mx + b
y = mx + b
y = 0x + 3
m=½
m=0
b=5
b=3
m = undefined
b = none
Example: Find the slope and
y-intercept of the equation.
d) 2x – 3y = 18
Solve for y.
2x – 3y = 18
– 3y = 18 – 2x
y = -6 + 2/3 x
y = 2/3 x - 6
y = mx + b
m = 2/3
y = 2/3 x - 6
b=-6
Your Turn: Find the slope and
y-intercept of the equation.
1) 3y + 12 = 6x
Solve for y.
3y = 6x – 12
2) 3y – x = 12
Solve for y.
y = 2x – 4
3y = x + 12
y = 1/3 x +4
y = mx + b
y = mx + b
y = 2x – 4
y = 1/3 x + 4
m=2
m = 1/3
b = –4
b=4
Your Turn:
1. y = -x + 8
m = -1 b = 8
2. x + 4y = 24
m = -1/4 b = 6
3. y = -1
4. x = 10
m = 0 b = -1
m = undef.
b = none
Example:
Write the equation of the line in slope-intercept form.
slope = ;
y-intercept = 4
y = mx + b
y=
x+4
Substitute the given values for m and b.
Simplify if necessary.
Example:
Write the equation that describes the line in slope-intercept form.
slope = –9; y-intercept =
y = mx + b
Substitute the given values for m and b.
y = –9x +
Simplify if necessary.
Your Turn:
Write the equation that describes the line in slope-intercept form.
slope = 3; y-intercept =
y = mx + b
Substitute the given values for m and b.
Simplify if necessary.
Your Turn:
Write the equation that describes the line in slope-intercept form.
slope =
; y-intercept = –6
y = mx + b
Substitute the given values for m and b.
Simplify if necessary.
Example: Writing Linear
Equations in Slope-Intercept Form
Write the equation that describes the line in slope-intercept form.
slope = 2; (3, 4) is on the line
Step 1 Find the y-intercept.
y = mx + b
4 = 2(3) + b
4=6+b
–6 –6
–2 = b
Write the slope-intercept form.
Substitute 2 for m, 3 for x, and 4 for y.
Solve for b. Since 6 is added to b,
subtract 6 from both sides to undo
the addition.
Example: Continued
slope = 2; b = - 2
Step 2 Write the equation.
y = mx + b
Write the slope-intercept form.
y = 2x + (–2)
Substitute 2 for m, and –2 for b.
y = 2x – 2
Your Turn:
A line has a slope of 8 and (3, –1) is on the line. Write the
equation that describes this line in slope-intercept form.
Step 1 Find the y-intercept.
y = mx + b
–1 = 8(3) + b
–1 = 24 + b
–24 –24
–25 = b
Write the slope-intercept form.
Substitute 8 for m, 3 for x, and –1 for y.
Solve for b. Since 24 is added to b,
subtract 24 from both sides to undo
the addition.
Continued
slope = 8; b = - 25
Step 2 Write the equation.
y = mx + b
Write the slope-intercept form.
y = 8x + (–25)
Substitute 8 for m, and –25 for b.
y = 8x – 25
Example: Writing an Equation
From a Graph
Find the linear equation of the line shown on the graph.
y-intercept = 4
y
Rise = –2
Step 1 Pick two points on the line and
determine the slope of the line from the
graph.
•
•
•
•
Run = 5
m=
Step 2 Find the y-intercept from the graph.
b=4
Step 3 Write the linear equation in slope-intercept form (y = mx+ b).
2
y x4
5
Your Turn:
Write the equation of the line shown.
B.
A.
y = 2x – 2
y = –x + 2
Example: Write an Equation
From Two Points
What equation in slope-intercept form represents the line that
passes through the points (2, 1) and (5, -8).
– Step 1 Use the two points to find the slope.
m
y2  y1 (8)  (1) 9


 3
x2  x1 (5)  (2)
3
– Step 2 Use the slope and one of the points to find b.
Use slope-intercept form
y  mx  b
Substitute -3 for m, 2 for x, and 1 for y
1  3(2)  b
Slove for b
7b
– Step 3 Substitute the slope and y-intercept into the slope-intercept
form (y = mx + b).
y 3x  7
Your Turn:
What equation in slope-intercept form represents the line that
passes through the points (1, -6) and (-3, 10).
– Step 1 Use the two points to find the slope.
m
y2  y1 (10)  (6) 16


 4
x2  x1
(3)  (1) 4
– Step 2 Use the slope and one of the points to find b.
Use slope-intercept form
y  mx  b
Substitute -4 for m, 1 for x, and -6 for y
 6  4(1)  b
Slove for b
2b
– Step 3 Substitute the slope and y-intercept into the slope-intercept
form (y = mx + b).
y  4 x  2
Your Turn:
What equation in slope-intercept form represents the line that
passes through the points (3, -2) and (1, -3).
– Step 1 Use the two points to find the slope.
m
y2  y1 (3)  (2) 1 1



x2  x1
(1)  (3)
2 2
– Step 2 Use the slope and one of the points to find b.
Use slope-intercept form
1
Substitute for m, 1 for x, and -3 for y
2
Slove for b
y  mx  b
1
 3  (1)  b
7 2
 b
2
– Step 3 Substitute the slope and y-intercept into the slope-intercept
form (y = mx + b).
1 7
y  x
2 2
Joke Time
• Why shouldn’t you write with a broken pencil?
• Because it is pointless!
• What do you call a man with no arms and no legs
playing in the leaves?
• Russell!
• Which side of a cheetah has the most spots?
• The outside.
Assignment
• 5-3 Part 1 Exercises Pg. 332 - 333: #6 – 40
even