Algebra 2
HS Mathematics
Unit: 04 Lesson: 01
©2010, TESCCC 08/01/10
Characteristics of Linear Functions (pp. 1 of 8)
Parent Function
Table
Linear Parent Function:
x y
y =
Domain:
Range:
What patterns do you observe in the table and graph of the linear parent function?
Review: Characteristics
Characteristics of linear functions (such as slope, intercepts, and
equations) have been addressed in previous math courses.
Forms of Linear Equations
Standard Form:
Point-Slope Form:
Slope-Intercept Form:
Slope-Intercept Form
In the slope-intercept form, explain how changes in m (the slope) and b (the y-intercept) affect the
graph of the parent function.
Changes caused by m Changes caused by b
Algebra 2
HS Mathematics
Unit: 04 Lesson: 01
©2010, TESCCC 08/01/10
Characteristics of Linear Functions (pp. 2 of 8)
Finding Slope
How would you define slope
?
Explain how to find slope from each representation.
Graph Table Two Points Equation
x y
2 5
4 19
7 40
(-1,3)
and
(4, -5)
A)
42 xy
B)
xy
3
2
5
C) 83
yx
Count
Find
Compute
Look
Finding Intercepts
What is an intercept
?
Explain how to find a
y-intercept from each representation.
From the graph: Determine where the line would cross the _____________
From a table: Look for the point _____________
From the bmxy equation:
Identify the _____________
From any equation: Plug in _____________, then solve for _____________
Explain how to find an
x-intercept from each representation.
From the graph: Determine where the line would cross the _____________
From a table: Look for the point _____________
From an equation: Plug in _____________, then solve for _____________
Algebra 2
HS Mathematics
Unit: 04 Lesson: 01
©2010, TESCCC 08/01/10
Characteristics of Linear Functions (pp. 3 of 8)
Finding Equations
There are several ways to determine the equation of a line, depending on the given information.
If you have… Then plug values into…
The slope and y-intercept
The slope, and coordinates for a point
Coordinates for two points
A table of data
Special Lines and Slopes
Lines Slopes Sample Equation(s)
Horizontal Lines
Vertical Lines
Parallel Lines
Perpendicular Lines
Sample Problems
Provide information about each linear function.
1.
x y
A) Fill in the table of values.
B) Determine the slope.
C) Find the equation for the line.
D) What are the intercepts?
Algebra 2
HS Mathematics
Unit: 04 Lesson: 01
©2010, TESCCC 08/01/10
Characteristics of Linear Functions (pp. 4 of 8)
2.
x y
-1 10
1 4
3 -2
5 -8
A) Graph the function.
B) Determine the slope.
C) Find the equation for the line.
D) What are the intercepts?
3.
x y
Equation: 4
3
2
xy
A) Complete a table of values.
B) Graph the function.
C) Write down the slope.
D) What are the intercepts?
Algebra 2
HS Mathematics
Unit: 04 Lesson: 01
©2010, TESCCC 08/01/10
Characteristics of Linear Functions (pp. 5 of 8)
From the given information, find the equation of the line both in slope-intercept and standard forms.
4. The slope of –
2
1
and contains point (–2, 5)
5. Contains points (2, -3) and (-6, 1)
6. Contains point (0, 4) and is parallel to y = 2x - 3
7. Contains point (-4, 5) and is perpendicular to 2x + 3y = 7
8. Contains (5, 1) and is perpendicular to y = 3.
9. Contains (5, 1) and is parallel to y = 3.
10. Contains (-2, -7) and is perpendicular to x = 4.
Algebra 2
HS Mathematics
Unit: 04 Lesson: 01
©2010, TESCCC 08/01/10
Characteristics of Linear Functions (pp. 6 of 8)
Practice Problems
1.
x y
A) Fill in the table of values.
B) Determine the slope.
C) Find the equation for the line.
D) What are the intercepts?
2.
x y
-6 3
-3 4
3 6
6 7
A) Graph the function.
B) Determine the slope.
C) Find the equation for the line.
D) What are the intercepts?
Algebra 2
HS Mathematics
Unit: 04 Lesson: 01
©2010, TESCCC 08/01/10
Characteristics of Linear Functions (pp. 7 of 8)
3.
x y
Equation: 5
2
3
xy
A) Complete a table of values.
B) Graph the function.
C) Write down the slope.
D) What are the intercepts?
Given information, find the equation of the line in y-intercept and standard forms.
4. The slope of –3 and contains the point (0, 3)
5. The slope of
3
5
and contains point (-6, -2)
6. Contains points (4, -1) and (-2, -13)
7. Contains point (-1, 2) and is parallel to x – 2y = -3
8. Contains point (5, -3) and is perpendicular to y = 5x - 4
9. Contains (-4, 3) and is perpendicular to y = 2.
10. Contains (8, -1) and is parallel to y = 1.
11. Contains (-2, -3) and is perpendicular to x = 2.
Algebra 2
HS Mathematics
Unit: 04 Lesson: 01
©2010, TESCCC 08/01/10
Characteristics of Linear Functions (pp. 8 of 8)
12. For each of the linear functions on the graph below, compare it to the linear parent function in
terms of vertical shifts and vertical compressions. Identify the parameter that determines the
change and determine the function rule.
Graph Transformations/Changes Equations
A) A)
B) B)
A) A)
B) B)
Algebra 2
HS Mathematics
Unit: 04 Lesson: 01
©2010, TESCCC 08/01/10
Characteristics of Linear Functions (pp. 1 of 8) KEY
Parent Function
Table
Linear Parent Function:
x y
-3.5 -3.5
-1 -1
0 0
1 1
4.5 4.5
6 6
y = x
Domain:
All real numbers (x )
Range:
All real numbers (y
)
What patterns do you observe in the table and graph of the linear parent function?
For each point, the x-values and y-values are equal. As x increases, y increases.
Fractional values are possible.
Review: Characteristics
Characteristics of linear functions (such as slope, intercepts, and
equations) have been addressed in previous math courses.
Forms of Linear Equations
Standard Form: ax + by = c (No fractions and a is positive value)
Point-Slope Form: y – y
1
= m(x – x
1
)
Slope-Intercept Form: y = mx + b
Slope-Intercept Form
In the slope-intercept form, explain how changes in m (the slope) and b (the y-intercept) affect the
graph of the parent function.
Changes caused by m Changes caused by b
m = 1, no change in slope of parent function
m > 0, m is positive and the line increases
from left to right
m < 0, m is negative and the line reflects in x-
axis; decreases from left to right
m > 1, slope of the parent function increases
(vertical stretch)
-1 < m < 1, excluding zero, slope of the parent
function decreases (vertical compression)
b = 0, no change in y-intercept of parent
function, stays at (0, 0)
b > 0, vertical shift up “b” units
b < 0, vertical shift down “b” units
Algebra 2
HS Mathematics
Unit: 04 Lesson: 01
©2010, TESCCC 08/01/10
Characteristics of Linear Functions (pp. 2 of 8) KEY
Finding Slope
How would you define slope
?
A line’s rate of change
A measure of a line’s “steepness” or “slant”
Explain how to find slope from each representation.
Graph Table Two Points Equation
x y
2 5
4 19
7 40
(here, m = 7)
(-1,3)
and
(4, -5)
(here, m =
8
5
)
A)
42 xy
(here, m = -2)
B)
xy
3
2
5
(here, m =
2
3
)
C)
83 yx
(*here, m = -3)
Count:
Rise
Run
(here, m =
4
2
= 2)
Find
y
or
change
in y
x
change
in x
Compute
y
2
– y
1
x
2
– x
1
Look
For the coefficient of
the ‘x’ term (as long
as the equation is in
y=mx+b form*)
Finding Intercepts
What is an intercept
?
Point where a line intersects (or crosses) the x-axis or y-axis
Explain how to find a
y-intercept from each representation.
From the graph: Determine where the line would cross the y-axis
From a table: Look for the point (0, y)
From the bmxy equation:
Identify the b-value
From any equation: Plug in x = 0, then solve for y
Explain how to find an
x-intercept from each representation.
From the graph: Determine where the line would cross the x-axis
From a table: Look for the point (x, 0)
From an equation: Plug in y = 0, then solve for x
Algebra 2
HS Mathematics
Unit: 04 Lesson: 01
©2010, TESCCC 08/01/10
Characteristics of Linear Functions (pp. 3 of 8) KEY
Finding Equations
There are several ways to determine the equation of a line, depending on the given information.
If you have… Then plug values into…
The slope and y-intercept Slope-intercept Form : y = mx + b
The slope, and coordinates for a point Point-slope Form : y – y
1
= m(x – x
1
)
Coordinates for two points
y
2
– y
1
then
y – y
1
= m(x – x
1
)
x
2
– x
1
A table of data A calculator’s lists (to do a linear regression)
Special Lines and Slopes
Lines Slopes Sample Equation(s)
Horizontal Lines A horizontal line has a slope of zero y = #
Vertical Lines A vertical line has undefined slope x = #
Parallel Lines
Slope of parallel lines are equal
(m
1
= m
2
)
Like y = 2x + 3 and y = 2x – 7
Perpendicular Lines
Slopes of perpendicular lines are
opposite reciprocals (
1
1
2
m
m)
Like y = 2x + 3 and y = -
1
2
x – 7
Sample Problems
Provide information about each linear function.
1.
x y
0 -5
7 9
4 3
-2 -9
A) Fill in the table of values.
B) Determine the slope.
m = 2
C) Find the equation for the line.
y = 2x – 5
D) What are the intercepts?
x-intercept: (2.5, 0)
y-intercept: (0, -5)
Algebra 2
HS Mathematics
Unit: 04 Lesson: 01
©2010, TESCCC 08/01/10
Characteristics of Linear Functions (pp. 4 of 8) KEY
2.
x y
-1 10
1 4
3 -2
5 -8
A) Graph the function.
B) Determine the slope.
m = -3
C) Find the equation for the line.
y = -3x + 7
D) What are the intercepts?
x-intercept: (
7
3
, 0)
y
-intercept: (0, 7)
3.
x y
-6 8
-3 6
3 2
6 0
Equation: 4
3
2
xy
A) Complete a table of values.
B) Graph the function.
C) Write down the slope.
m = -
2
3
D) What are the intercepts?
x-intercept: (6, 0)
y-intercept: (0, 4)
Algebra 2
HS Mathematics
Unit: 04 Lesson: 01
©2010, TESCCC 08/01/10
Characteristics of Linear Functions (pp. 5 of 8) KEY
From the given information, find the equation of the line both in slope-intercept and standard forms.
4. The slope of –
2
1
and contains point (–2, 5)
y = -(
1
2
)
x + 4
x + 2y = 8
5. Contains points (2, -3) and (-6, 1)
y = -(
1
2
)
x - 2
x + 2y = -4
6. Contains point (0, 4) and is parallel to
y = 2x - 3
y = 2x + 4
2
x – y = -4
7. Contains point (-4, 5) and is perpendicular to 2
x + 3y = 7
y =
3
2
x + 11
3
x – 2y = -22
8. Contains (5, 1) and is perpendicular to
y = 3.
y = 3 is horizontal, so perpendicular line is vertical with an “x = #” equation
x = 5 (no slope-intercept form)
9. Contains (5, 1) and is parallel to
y = 3.
y = 3 is horizontal, so parallel line is also horizontal with a “y = #” equation
y = 1
10. Contains (-2, -7) and is perpendicular to
x = 4.
x = 4 is vertical, so perpendicular line is horizontal with a “y = #” equation
y = -7
Algebra 2
HS Mathematics
Unit: 04 Lesson: 01
©2010, TESCCC 08/01/10
Characteristics of Linear Functions (pp. 6 of 8) KEY
Practice Problems
1.
x y
0 1
-3 3
-6 5
3 -1
A) Fill in the table of values.
B) Determine the slope.
m =
2
3
C) Find the equation for the line.
y = -
2
3
x + 1
D) What are the intercepts?
x-intercept: (1.5, 0)
y-intercept: (0, 1)
2.
x y
-6 3
-3 4
3 6
6 7
A) Graph the function.
B) Determine the slope.
m =
1
3
C) Find the equation for the line.
y =
1
3
x + 5
D) What are the intercepts?
x-intercept: (-15, 0)
y-intercept: (0, 5)
Algebra 2
HS Mathematics
Unit: 04 Lesson: 01
©2010, TESCCC 08/01/10
Characteristics of Linear Functions (pp. 7 of 8) KEY
3.
x y
-2 -8
0 -5
2 -2
4 1
Equation:
5
2
3
xy
A) Complete a table of values.
B) Graph the function.
C) Write down the slope.
m =
3
2
D) What are the intercepts?
x-intercept: (10/3, 0)
y-intercept: (0, 5)
From the given information, find the equation of the line both in slope-intercept and standard forms.
4. The slope of –3 and contains the point (0, 3) y = -3x + 3, 3x + y = 3
5. The slope of
3
5
and contains point (-6, -2) y =
5
3
x + 8, 5x - 3y = -24
6. Contains points (4, -1) and (-2, -13) y = 2x - 9, 2x - y = 9
7. Contains point (-1, 2) and is parallel to x – 2y = -3 y =
1
2
x +
5
2
, x - 2y = -5
8. Contains point (5, -3) and is perpendicular to y = 5x – 4 y = -
1
5
x - 2, x + 5y = -10
9. Contains (-4, 3) and is perpendicular to y = 2 x = -4
10. Contains (8, -1) and is parallel to y = 1 y = -1
11. Contains (-2, -3) and is perpendicular to x = 2 y = -3
Algebra 2
HS Mathematics
Unit: 04 Lesson: 01
©2010, TESCCC 08/01/10
Characteristics of Linear Functions (pp. 8 of 8) KEY
12. For each of the linear functions on the graphs below, compare it to the linear parent function in
terms of vertical shifts and vertical compressions. Identify the parameter that determines the
change and determine the function rule.
Graph Transformations/Changes Equations
A)
Vertical stretch by a factor of 3, m = 3
A)
y = 3x
B)
Vertical compression by a factor of
1
3
,
m =
1
3
B)
y =
1
3
x
A)
Vertical shift up by 6 units, b = 6
A)
y = x + 6
B)
Vertical shift down by 6 units, b = -6
B)
y = x - 6
Reflection over the x-axis, m = -1
y = -x
