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MATH 1314 College Algebra
Matrices
Matrices
Section Notes:
• A matrix is a __________
rectangular _____
array of numbers.
• The numbers in a matrix are called ______
terms or _______.
entries
• An __________
augmented matrix is a matrix that represents a system
of equations.
Gaussian Method uses these matrices to solve a
• The _________
system of linear equations.(Review in textbook.)
row operations to eliminate any entry
• This method relies on ____________
in the matrix.
• Only 3 Row Operations are used on a matrix:
Switch any two rows.
 ________
Multiply any row by a nonzero constant.
 ________
Add any two rows, …or add a non-zero constant
 ________
multiple of one row to another.
Matrices
• A matrix in Row Echelon Form has:
main diagonal entries of 1 from upper left to lower right
with 0 entries below each main diagonal entry.
(The last entry on main diagonal may also be 0.)
1
0
0
1
1
0
1 2
3 7
1 0
Matrices
• A matrix in Row Echelon Form can show that the
system of linear equations has:
only one solution
infinitely many solutions
or no solution.
1
0
0
𝒙
1
1
0
𝒚
1
1
−4 10
1 −2
𝒛 = 𝒄
1 −5 0 1
0 1 3 −7
0 0 0 0
1
0
0
0
1
0
1
1
−4 10
0 −2
P
1𝑧 = −2
𝑧 = −2
P
0𝑧 = 0
0=0
O
0𝑧 = −2
…here’s how you can tell…
 matrix is in Row Echelon Form.(read in textbook)
 last row shows one, many, or no solution exists.
Matrices
Matrix in Row Echelon Form.
1
– 0
0
1
1
0
1 2
3 7
1 5
– Does this matrix have:
Pa)
Only one solution
b) Infinitely many solutions
c) or No solution
Matrices
• Writing the solution set for Infinitely Many Solutions:
1 −5 0 1 → 𝑥 − 5𝑦 = 1
– Example: 0 1 3 −7 → 𝑦 + 3𝑧 = −7
0 0 0 0
– Solve Row2 equation for y. (Solutions will be in terms of 𝑧.)
→ 𝑦 + 3𝑧 = −7
𝑦 = −3𝑧 − 7
– Then solve Row1 Equation for x and simplify.
You may need to substitute y.
→ 𝑥 − 5𝑦 = 1
𝑥 = 5𝑦 + 1
𝑥 = 5 −3𝑧 − 7 + 1
𝑥 = −15𝑧 − 35 + 1
𝑥 = −15𝑧 − 34
Matrices
HW practice
• Solve the system using Gaussian Elimination.
Eqt1
2𝑥 + 𝑦 − 𝑧 = 3
Eqt2
−𝑥 + 2𝑦 + 4𝑧 = −3
Eqt3
𝑥 − 2𝑦 − 3𝑧 = 4
• First, rewrite as an augmented matrix.
𝑥
C1
R1
R2
R3
2
−1
1
𝑦
C2
𝑧
C3
1
2
−2
−1
4
−3
constants
C4
3
−3
4
Matrices
• Use Row Operations to solve.
Need a 1 in R1C1.
2
1 −1 3
Try switching R1 and R3
−1 2
4 −3
1 −2 −3 4
1 −2 −3 4
• −1 2
4 −3
These two entries must
be eliminated by using
2
1 −1 3
R1 and opposites.
Matrices
•
•
1 −2 −3
−1 2
4
2
1 −1
1
0
2
−2 −3
0
1
1 −1
4
−3
3
4
1
3
Eliminate -1 with its opposite in R1.
Use R2 + R1 to replace R2.
R2
-1 2 4 -3
R1 + 1 -2 -3 4
0 0 1 1 New R2
Eliminate 2 with its opposite using R1.
Use R3 + (-2)R1 to replace R3.
R3
2 1 -1 3
(-2)R1 + -2 4 6 -8
0 5 5 -5 New R3
•
1
0
0
−2 −3
0
1
5
5
4
1
−5
C1 is complete, R1 no longer used.
Matrices
1 −2
• 0 0
0 5
1 −2
• 0 0
0 1
1 −2
• 0 1
0 0
−3
1
5
−3
1
1
−3
1
1
4
1
−5
4
1
−1
4
−1
1
Need a 1 in R2C2.
First try to multiply (1/5)R3.
(1/5)R3
0
1
1 -1
New R3
Then switch R2 and R3.
Matrix is in Row Echelon Form.
Find solution for z in R3.
Then use back-substitution to find
solution for y and x.
Variable coefficients
•
Matrices
1𝒙 −2𝒚 −3𝒛 = 4 → 𝑥 − 2𝑦 − 3𝑧 = 4
0 𝒙 1𝒚 1𝒛 = −1 → 𝑦 + 𝑧 = −1
0𝒙 0𝒚 1𝒛 = 1 → 𝑧 = 1
• First solution value: 𝑧 = 1
• Back-substitute z into R2 to find y-value:
→ 𝑦 + 𝑧 = −1 → 𝑦 + (1) = −1
→ 𝑦 = −2
• Back-substitute z and y in R1 to find x-value:
→ 𝑥 − 2𝑦 − 3𝑧 = 4
→ 𝑥 − 2(−2) − 3(1) = 4
→ 𝑥+4−3=4
→ 𝑥=3
Matrices
HW practice
So, the System of Linear Equations below
2𝑥 + 𝑦 − 𝑧 = 3
−𝑥 + 2𝑦 + 4𝑧 = −3
𝑥 − 2𝑦 − 3𝑧 = 4
has only One Solution, and the solution is:
𝑥 = 3, 𝑦 = −2, 𝑧 = 1
Substitute the solutions into each equation to
verify they are correct.
Matrices
• Translating a written application problem to
equations:
1. Read carefully.(read two or more times)
2. Identify and label variables and number of
equations.
 variables: The last question gives you a hint.
 equations: How many quantity totals are stated?
3. Form system of equations from variables and
quantity totals.
Matrices
• Section 8.2: Exercise #81-College Algebra e-text
1. Read.
2. Variables and Totals:
 Variables: Delta x, Beta y, Sigma z
 Totals(equations): painting, drying, polishing
Matrices
• Section 8.2: Exercise #81-College Algebra e-text
3. Form equations:
•
•
•
Painting hours
Drying hours
Polishing hours
Delta
𝑥
Beta
𝑦
Sigma
𝑧
Total
hours
10𝑥 + 16𝑦 + 8𝑧 = 240
3𝑥 + 5𝑦 + 2𝑧 = 69
2𝑥 + 3𝑦 + 𝑧 = 41
Now form augmented matrix to solve.
MATH 1314 College Algebra
Unit 1
Functions and their Graphs
Section: 3.1
Functions
Section :_____
Reading assignment-Terminology. Fill in the blank correctly.
• A relation is:
– a _____________ between two sets X and Y.
correspondence ? or corollary ?
• A function from set X into set Y is:
– a relation in which each 𝑥-value is associated
with ___________ one 𝑦-value.
more than? or exactly ?
– The notation for ordered pairs is ______.
(𝑥, 𝑦)? or 𝑓(𝑥) ?
• Fill in the blank with the correct term below.
(range input dependent domain independent output)
input
domain
independent
– 𝑥-variable: ____________,
____________,
_____________
dependent
output
range
– 𝑦-variable: ____________,
____________,
_____________
• Interval Notation for domain:
– parentheses mean “not equal to”, brackets mean “equal to”.
Functions
Section :_____
• Relation or Function?(Refer to definitions.)
2)
1)
Relation
Function
X
Y
X
Y
-1
5
8
3
1
-6
4
0
-9
7
-2
3
Each 𝑥-value associated
with exactly one 𝑦-value.
One 𝑥-value associated with
two different 𝑦-values.
4)
3)
Function
Relation
{ (-4, 0), (1, 9), (0, -6), (-8, 3), (1, -2) } { (2, 1), (3, 0), (-4, 6), (1, -3), (0, 5) }
One 𝑥-value associated with
two different 𝑦-values.
Each 𝑥-value associated
with exactly one 𝑦-value.
Functions Section :_____
• Is the equation below a function of 𝑥?
2) −𝑥 2 + 𝑥 + 6 + 𝑦 = 0
1) 𝑥 2 + 𝑦 2 = 9
 Solve for 𝑦.
 Solve for 𝑦.
𝑦2 = 9 − 𝑥2
𝑦2 = ± 9 − 𝑥2
𝑦 = 𝑥2 − 𝑥 − 6
 Only one equation for 𝑦.
𝑦 = ± 9 − 𝑥2
𝑦 = 9 − 𝑥 2 or 𝑦 = − 9 − 𝑥 2
 Two different equations for 𝑦.
𝑥 2 + 𝑦 2 = 9 is not a
function of 𝑥.
−𝑥 2 + 𝑥 + 6 + 𝑦 = 0
is a function of 𝑥 .
Functions
Section :_____
• Function Notation 𝑦 = 𝑓(𝑥): “the value of 𝑓 at 𝑥”.
– To find the value of 𝑓(𝑥), substitute 𝑥-value into 𝑓(𝑥) and simplify.
Example: Find the following for 𝑓 𝑥 = 𝑥 2 + 2𝑥.
a) 𝑓(3)
b) 𝑓 𝑥 + 𝑟
𝑓 𝑥 = 𝑥 2 + 2𝑥
𝑓 𝑥 = 𝑥 2 + 2𝑥
𝑓 𝑥 + 𝑟 = 𝑥 + 𝑟 2 + 2(𝑥 + 𝑟)
𝑓 3 = 3 2 + 2(3)
= 𝑥 + 𝑟 𝑥 + 𝑟 + 2(𝑥 + 𝑟)
= 9 + 6 = 15
= 15
= (𝑥 2 + 𝑥𝑟 + 𝑥𝑟 + 𝑟 2 ) + 2𝑥 + 2𝑟
= 𝑥 2 + 2𝑥𝑟 + 𝑟 2 + 2𝑥 + 2𝑟
Refer to page 210 for other examples.
P
• Domain of 𝑓(𝑥) is the set of all real numbers 𝑥 that define 𝑓,
except 𝑥-values that:
– cause division by zero
– cause even roots of negative numbers.
P
Functions
Section :_____
𝑥+3
• Find the domain of 𝑓 𝑥 =
. (Write the domain using
𝑥−5
Denominator may need to
Interval Notation.)
1). Find restricted 𝑥-value(s) of function.
Cannot divide by zero, so
𝑥−5≠0
Restricted value is
𝑥≠5
2). Cross out restricted value(s)
−∞
on number line.
3). Write domain interval(s)
from left to right.
be factored in order to
solve for restricted values
in other related problems,
producing two or more
restricted values.
∞
0
5
−∞
∞
0
5
−∞, 5 ∪ (5, ∞)
P
Functions
Section :_____
• Find the domain of 𝑓 𝑥 = 6𝑥 + 18 .
(Write the domain using Interval Notation.)
1). Find restricted 𝑥-values of function.
6𝑥 + 18 is not real if 6𝑥 + 18 is negative, so 6𝑥 + 18 ≥ 0
Solve for x:
6𝑥 ≥ −18
𝑥 ≥ −3
All x-values are restricted except 𝑥 ≥ −3.
2). Cross out restricted value(s)
on number line.
−∞
3). Write domain interval(s)
from left to right.
−∞
∞
-3
-3
0
∞
0
[−3, ∞)
P
Functions
Section :_____
• Use the previous two Practice Exercises to find the
𝑥
domain. 𝑓 𝑥 =
𝑥−2
1). Find restricted 𝑥-values of function.
First, numerator 𝑥 requires that 𝑥 ≥ 0.
Second, denominator requires that 𝑥 − 2 ≠ 0, so 𝑥 ≠ 2.
Together, this means 𝑥 ≥ 0, but 𝑥 ≠ 2.
2). Cross out restricted value(s)
on number line.
−∞
3). Write domain interval(s)
from left to right.
−∞
∞
0
2
∞
0
2
[0,2) ∪ (2, ∞)
P
MATH 1314 College Algebra
Functions and their Graphs
Section: 3.2
Graphs of Functions
Section :____
• A function can be identified from its graph by
Vertical Line
the ____________Test.
• The Vertical Line Test states: if a vertical line
intersects a graph _____________,
more than once the graph
is ___________________.
not a graph of a function
• Is this the graph of a function?
A vertical line intersects
the graph only once.
YES!
A vertical line intersects
the graph more than once.
NO!
Graphs of Functions
Section :____
• Information on a function from its graph:
 𝑦 = 𝑓(𝑥), so (𝑥, 𝑦) can also be (𝑥, 𝑓(𝑥))
 To find 𝑓(𝑐) on graph, look for 𝑦 on graph when 𝑥 = 𝑐.
The 𝑦-value is 𝑓(𝑐). (The value 𝑐 is a real number.)
 To find 𝑥 when 𝑓 𝑥 = 𝑏, look for 𝑥 on graph when 𝑦 = 𝑏.
(The value 𝑏 is a real number.)
 Domain of graph: domain is set of all 𝑥-values across graph.
(Always read domain from graph left to right.)
 Range of graph: range is set of all 𝑦-values across graph.
(Always read range from graph bottom to top.)
 𝑥-intercepts: (𝑥, 0) Points where graph intersects 𝑥-axis.
 𝑦-intercepts: (0, 𝑦) Points where graph intersects 𝑦-axis.
Graphs of Functions
Section :___
• Information from the graph of a function.
– a) Find 𝑓(−21).
What−6
is 𝑦 when 𝑥 is −21?
Answer:
– b) For what numbers 𝑥
is 𝑓(𝑥) = 0?
Answer: −18, −3, 12
– c) For what numbers 𝑥
is 𝑓(𝑥) = 6?
Answer: −6, 18
– d) What is the domain of 𝑓?
Answer: [−21,18]
– e) What is the range?
Answer: [−6,9]
Graphs of Functions
Section :___
• Information from a function about its graph.
If 𝑓 𝑥 = 2𝑥 2 − 𝑥 − 1:
Equation Editor
a) Is point (2,5) on graph of 𝑓(𝑥)? Yes
Only if 𝑦 = 5, when 𝑥 = 2.
𝑓 𝑥 = 𝑦 = 2𝑥 2 − 𝑥 − 1
𝑦 = 2(2)2 − 2 − 1
𝑦=5
P
TI-83/84 : 1) Press Y=.
Enter 𝑓(𝑥) in Equation Editor.
2) Press 2nd GRAPH for TABLE.
3) Look for 𝑦 when 𝑥 = 2.
X-variable
Exponent
Graphs of Functions
Section :___
• Information from a function about its graph.
If 𝑓 𝑥 = 2𝑥 2 − 𝑥 − 1:
b) If 𝑥 = −3, what is 𝑓(𝑥)? 20
𝑓 𝑥 = 2𝑥 2 − 𝑥 − 1
𝑓(−3) = 2(−3)2 − −3 − 1
= 20
P
 With this information,
list a point on graph of 𝑓.
(−3,20)
Graphs of Functions
Section :___
• 𝑓 𝑥 = 2𝑥 2 − 𝑥 − 1:
c) If 𝑓(𝑥) = −1, what is 𝑥?
What point(s) are on graph of 𝑓?
𝑥 = 0,
1
2
1
( 0, −1 ),
2
1
( , −1 )
2
Set 𝑓 𝑥 = −1, solve for 𝑥
𝑓 𝑥 = 2𝑥 2 − 𝑥 − 1 = −1 When solving Quadratic equations,
always write in Standard
Form first.
2
2
Standard Form: 𝑎𝑥 + 𝑏𝑥 + 𝑐 = 0
2𝑥 − 𝑥 = 0
𝑥 2𝑥 − 1 = 0
𝑥 = 0, 2𝑥 − 1 = 0
𝑥 = 0, 𝑥 =
1
2
P
Graphs of Functions
Section :___
• 𝑓 𝑥 = 2𝑥 2 − 𝑥 − 1:
d) What is the domain of 𝑓(𝑥)?
−∞, ∞
1) Does 𝑓 𝑥 show possible division by zero,
or possible square roots of negative values?
No, so 𝑓(𝑥) can use all real numbers.
P
Graphs of Functions
Section :___
• If 𝑓 𝑥 = 2𝑥 2 − 𝑥 − 1:
1
− ,1
2
e) What x-intercept(s), if any,
are on graph of 𝑓?
x-intercept is 𝑥, 0 , so…
Factoring is needed to solve
𝑓 𝑥 = 2𝑥 2 − 𝑥 − 1 = 0
the Quadratic equation.
(2𝑥 + 1 )( 𝑥 − 1 ) = 0
2𝑥 + 1 = 0,
𝑥−1=0
𝑥=−
1
2
𝑥=1
 f) What y-intercept, if any,
is on graph of 𝑓?
P
−1
y-intercept is 0, 𝑦 , so… 𝑓 0 = 2(0)2 −(0) − 1 =
MATH 1314 College Algebra
Unit 1
Library of Functions
Section: 3.4
Library of Functions
Section :____
• This section covers several basic functions to begin the main
study of this course.
• Study and learn all properties and graphs for the functions
listed in this section that you were assigned.
• The next section will cover a collection of techniques called
Transformations to graph functions similar to these basic
functions.
• After completing the work in this section,
your goal is to successfully:
– recognize and sketch the graph of each function,
– identify the Domain and Range of each function.
• If you do not meet this goal, you will need to review this
section until you can in order to prepare for Section 3.5.
Library of Functions
Section :____
• Quick Check!
∞
3
 From your Notes assignment:
 State the basic function
for the given graph shape.
Square Function
_____________________
−∞
(−∞, ∞)
 State it’s Domain: _________
[0, ∞)
 State it’s Range: ________
−3
3
−3
−∞
∞
Library of Functions
Section :____
• Quick Check!
∞
3
 From your Notes assignment:
 State the basic function
for the given graph shape.
Reciprocal Function
_____________________
−∞
−3
3
−3
−∞
(−∞, 0) ∪ (0, ∞)
 State it’s Domain: _________________
(−∞, 0) ∪ (0, ∞)
 State it’s Range: _________________
∞
Library of Functions
Section :____
• Quick Check!
∞
3
 From your Notes assignment:
 State the basic function
for the given graph shape.
Cube Root Function
_____________________
−∞
(−∞, ∞)
 State it’s Domain: _________
(−∞, ∞)
 State it’s Range: ________
−3
3
−3
−∞
∞
Library of Functions
Section :____
• Quick Check!
∞
3
 From your Notes assignment:
 State the basic function
for the given graph shape.
Identity Function
_____________________
−∞
(−∞, ∞)
 State it’s Domain: _________
(−∞, ∞)
 State it’s Range: ________
−3
3
−3
−∞
∞
Library of Functions
Section :____
• Piecewise-defined Function:
is a function defined using different equations on
different parts of its domain.
𝑥2
 Example: 𝑓 𝑥 =
𝑥−3
𝑓 𝑥 = 𝑥 2 is defined
only over this interval.
−∞
-5
𝑖𝑓 − 2 ≤ 𝑥 < 0
𝑖𝑓 𝑥 ≥ 0
𝑓 𝑥 = 𝑥 − 3 is defined
only over this interval.
∞
5
[ )[
5
-3
−∞
)
∞
P
Library of Functions
Section :____
• Piecewise-defined Function.
 Practice: 𝑓 𝑥 =
𝑥2
𝑥−3
𝑖𝑓 − 2 ≤ 𝑥 < 0
𝑖𝑓 𝑥 ≥ 0
 a) Find 𝑓(−1). 𝑥 = −1 lies in interval −2 ≤ 𝑥 < 0.
Find 𝑓(−1) using 𝑓 𝑥 = 𝑥 2 .
𝑓(−1) = (−1)2 = 1
P
𝑥 = 7 lies in interval 𝑥 ≥ 0.
 b) Find 𝑓 7 .
Find 𝑓(7) using 𝑓 𝑥 = 𝑥 − 3.
𝑓 7 = 7 −3= 4
P
Library of Functions
Section :____
Practice: Read domain interval for each
section of piecewise-defined
graph given.
y
7
Section I
x
-5
5
-3
Domain intervals:
Section I
Section II
[−5,2]
[2,4)
Section II
Library of Functions
Section :____
Practice: Write a definition for the
piecewise-defined graph given.
4
𝑓(𝑥) = − 3 𝑥 if −3 ≤ 𝑥 ≤
0
4
4
𝑚𝑥 + 𝑏 → − 𝑥 + 0 → − 𝑥
3
3
7
(−3, 4)
(5,2)
Domain for Left line segment is:
Left line segment is 𝑦 = 𝑚𝑥 + 𝑏.
Find slope 𝑚 and 𝑦-intercept.
y
−4
-5
P
(0,0)
3
5
-3
 Complete the second section in the same way.
𝑓(𝑥) =
2
𝑥 if
5
0 ≤𝑥≤
5
P
2
5
x
Review and Complete
HW 3.4-Library of Functions
MATH 1314 College Algebra
Unit 1
Transformations
Section: 3.5
Transformations
Section :____
• As stated in the textbook for today’s Lecture(Section 3.5),
techniques used to graph functions similar to basic functions
Transformations
are called _______________.
three
• How many different categories were outlined? ______
• These main categories of transformations were:
Reflections
 ______________
𝑥
if 𝑦 = −𝑓(𝑥), reflect 𝑓(𝑥) about the ___-axis
𝑦 -axis
if 𝑦 = 𝑓(−𝑥), reflect 𝑓(𝑥) about the ___
Stretch/Compress
 _____________________
vertically
if 𝑦 = 𝑎𝑓 𝑥 , 𝑎 > 0, stretch/compress 𝑓(𝑥) __________
if 𝑦 = 𝑓 𝑎𝑥 , 𝑎 > 0, stretch/compress 𝑓(𝑥) ____________
horizontally
Shifts
 ________
horizontally
if 𝑦 = 𝑓 𝑥 ± ℎ , ℎ > 0, shift 𝑓(𝑥) ____________
vertically
if 𝑦 = 𝑓(𝑥) ± 𝑘, 𝑘 > 0, shift 𝑓(𝑥) __________
 Review symmetry of graphs as needed(p165).
Transformations
Section :____
∞
 Prepare your mind for
Transformations:
In the following graph:
A (3,2)
B
1
−∞
-4
-1
∞
3
 if point A moves to point B,
is the x- or y-coordinate value
x-coordinate
transformed? _____________
−∞
horizontally
 why? …because point A shifted __________.
 what operation shifts point A to point B?
subtract 7 from x-coordinate
__________________________
(3,2)
 find the new coordinate for point A after
(3 − 7,2)
it moves to point B.
(−4,2)
P
Library of Functions
Section :____
• If a basic function graph changes from its initial
direction, shape, or position, then its function
has had one or more Transformations
_____________ applied.
 For the given graph shape:
∞
3
 Identify its initial basic function.
Square Root Function
____________________
∞
−∞
−3
3
 Explain how the initial
basic function graph has
−3
been Transformed?
−∞
Shifted right two units and shifted up one unit
________________________________________
 State the Domain and Range of the
[2, ∞) Range______
[1, ∞)
Transformed function: Domain ______
Transformations
Section :____
 Prepare your mind:
• If 𝑓 𝑥 = − 𝑥 is a transformed function,
𝑥
 Its basic function is: _______.
 The basic function graph is:
 Graph of 𝑓 𝑥 = − 𝑥 looks like:
(0,0)
(1,1)(4,2)
(0,0) (1, −1)
(4, −2)
 What effect does leading − sign have
reflects graph about 𝑥-axis
on basic graph? _________________________
multiply 𝑦-coordinates by −1
on its basic coordinates?________________________
Transformations
Section :____
 Prepare your mind:
• If 𝑓 𝑥 = (𝑥 + 3)2 −2 is a transformed function,
𝑥2
 Its basic function is: _______.
 The basic function graph is:
 The graph of 𝑓(𝑥) looks like:
 What effect does + 3 have on basic graph?
shifts graph left 3 units
________________________________
subtract 3 from 𝑥-coordinates
The coordinate rule is: ____________________________.
 What effect does − 2 have on basic graph?
shifts graph down 2 units
________________________________
subtract 2 from 𝑦-coordinates
The coordinate rule is: ____________________________.
Transformations
Section :____
 Local maximum or minimum on graph of 𝑓(𝑥):
• A point on section of graph that is higher or lower
than any other points around it.
Transformations
Section :____
Determining increasing, decreasing Intervals
 𝑓(𝑥) increases over an open interval of 𝑥-values if
graph only climbs from left-right over the interval.
 𝑓(𝑥) decreases over an open interval of 𝑥-values if
graph only falls from left-right over the interval.
 Open intervals only use ( ).
∞
Increasing Decreasing
Increasing
5
Local −1)
maximum (−1,3)
(−∞,
(3, ∞)
(−1, 𝑦)
−∞
∞
-5
5
-5
−∞
3, 𝑦
Local minimum
Transformations
Section :____
 To find local maximum on TI-83/84:
𝑓 𝑥 = 𝑥 3 − 4𝑥
• Enter equation in Y1 .
• Press 2nd TRACE
Select command :maximum
• Left Bound?: move cursor to left
side of maximum, press ENTER.
• Right Bound?: move cursor to
right side of maximum, press ENTER.
• Guess?: press ENTER.
 For local minimum use :minimum
Transformations
Section :____
 Practice: If 𝑓 𝑥 = 𝑥 3 − 16𝑥, for −7 < 𝑥 < 7
find the following:
a) Choose the correct graph
of 𝑓(𝑥) below.
PA
B
Before graphing, change
WINDOW settings
Xmin: −7
Xmax: 7.
(Adjust Ymin Ymax as needed.)
C
Transformations
Section :____
 Practice: If 𝑓 𝑥 = 𝑥 3 − 16𝑥, for −7 < 𝑥 < 7
find the following:
b) 𝑥-intercepts of 𝑓 𝑥
Solve for 𝑥.
(or use TI-83)
c) 𝑥-intercepts of 𝑓 𝑥 + 2
* 𝑓 𝑥 + 2 → shift 𝑓(𝑥) left 2 units
→ subtract 2 from 𝑥-values
𝑥 3 − 16𝑥 = 0
𝑥(𝑥 + 4)(𝑥 − 4) = 0
𝑥 = 0, 𝑥 + 4 = 0, 𝑥 − 4 = 0
𝑥 = 0, 𝑥 = −4,
𝑥=4
−4, 0, 4
P
−4 − 2
0−2
−6, −2, 2
4−2
P
End - Unit 1
Complete all Ch3
assignments in time to take
Ch3 Exam by Due Date.