The Daily Parabola
Featured story: ​
Inside Turn Left
Industries “Graphing Functions changed
my life” - Ratchel the Satchel
Included in this Issue:​
Variables!
​
Expressions! Functions! Krabby Patties!
and MORE!!
Bonus!! Special Real World
Problems!!! Super Fun!!
In next week’s issue: ​
Linear Programming, Polynomials, and Inverses of
Fractions!
by Jamie Kesten, Jia Seow, Philip Carey, Hannah Yang
Period 1
1-3 Variables and Expressions
Goal​
: Utilize Order of Operations to evaluate variable expressions
Definition​
:
Variable: A variable is a letter which stands for an unspecified number from a given set
Agreement​
: Unless otherwise specified, the domain of a variable will be assumed to be the
set of all real numbers
Definition:
Expression: An expression is a collection of variables and constants connected by operation
signs ( +, -, x, /, etc.) which stands for a number.
Definitions:
Subtraction: x - y means x + (-y)
Division: x/y means x * 1/y
Exponentiation: x^n means n, x’s multiplied together. For example, x³ means x*x*x
Agreement:
Order of Operations:
1. Do any operations inside parentheses first
2. Do any exponentiations next
3. Do multiplication and division in the order in which they occur, from left to right
4. Do addition and subtraction last, in the order in which they occur, from left to right
Definition:
Absolute Value: Distance between the number x and the origin of the number line
| x | = x if x is positive (or 0)
| x | = -x if x is negative
Key Points:
1. When doing a calculation, substitute the same value of x everywhere it appears
2. The answer you get correlates with the order of operations, so follow Order of
Operations!
Practice Problem: ​
6x² - 3x(x - 3y) + 3y(3y + x) - 3x²
Step 1: Factoring: 6x² - 3x² + 9xy + 9y² + 3xy - 3x²
Step 2: Combining Like Terms: 12xy + 9y²
Final Answer: 12xy + 9y²
Inside Turn Left Industries
Turn Left Industries is a company that fights for the normalization of
making 3 left turns to make a right turn. It was started by a high
school girl named Ratchel the Satchel who had to bike to school
everyday, she only knew how to make left turns which often led her to
go the wrong way and fall of her bike. Her company has collected lots
of data on the need for people to make 3 left turns instead of a right
turn and she has presented her data to the public, however it did not
have the expected impact that Ratchel hoped for. After viewing her
presentation, she has decided that her presentation was ineffective due to the fact that she
presented her information in just numbers instead of graphs. She decided to head back to Los
Altos High School and learn how to graph a function, and her new presentation was a success!
Ratchel the Satchel says, “Graphing functions changed my life! It really allows people to
finally see my vision!”
2-2 Graphs of Functions
Overview: Learning to graph a function when given the equation and understanding domain
and range.
Definition:
Domain: The domain of a function is the set of values of the independent variable.
Definition:
Range: The range of a function is the set of values of the dependent variable corresponding to
all values of the independent variable in the domain.
Method:
2​
1. Find the vertex using the equation −b
+ bx + c = y, which finds the
2a where ax​
x-coordinate of the vertex.
2. Plug in values for X and plot points
Tips:
1. Try to isolate the variable firsts
2​
2. If a in ax​
+ bx + c = y is positive the function will open upwards, and if a is negative
the function will open downwards.
Example Problems:
Plot the graph of the function and tell the range:
2​
1. y = 0.4x​
, domain = {real numbers}
Step 1: Find the vertex using
−b
2a
:
−0
2(.4)
2​
Step 2: Plug 0 in for x:
y = 0.4(0)​
Step 3: Plug in values for x:
x
y
=0
Vertex = (0,0)
2
1.6
1
.4
-1
-.4
Step 4: Plot points
Step 5: Find the range:
y≥ 0
Real World Application:
Fall Ing is trying to find how long it will take for him to get to work. He works at Turn Left
Factory which is owned by Ratchel the Satchel. He has 30 minutes to get to work. For every
9 cars on the road it will take him and extra 3 minutes to get work. Assume that it will take
him 15 minutes to get to work if there are no cars on the road.
a. Write the equation for how long it will take him to get to work, using the variables t
for the time it will take to get to work and c for the number of cars on the road.
b. Graph the function
c. If there are 130 cars on the road how long, will Fall Ing make it to work on time?
Solutions :
a. t = 13 c + 15
b.
c. Plug in 130 for c: t =
1 (130)+ 15
3
t = 43 13 + 15
t = 58 13
58 13 > 30
No he will be late.
Additional Practice:
1. Plot the graph of the function:
a. y = x+4, domain = {positive numbers}
b. y =
−1
x
, domain = {all real numbers}
4-6 Systems of Linear Equations with Three or More Variables
Goal: Be able to find the single ordered triple that satisfies a system of three linear equations
with three variable.
Key Points:
1. Eliminate one variable from the original system of three equations to get a system of
two equations with two variables
2. Solve for one of these variables through linear combination
3. Plug the value into the 2-system equation to find the value of the other variable
4. Use the 2 values to find the value of the third variable in an original equation
Ex: 7x + 5y - 3z = 16
3x - 5y + 2z = -8
5x + 3y - 7z = 0
Step 1: Get a 2 variable System through Linear Combination:
7x + 5y -3z = 16
3x - 5y + 2z = -8
+-----------------------
10x - z = 8
(3x - 5y + 2z = -8) x3 -------------> 9x - 15y + 6z = -24
(5x + 3y - 7z = 0) x5 --------------> 25x + 15y - 35z = 0
+---------------------34x - 29z = -24
Step 2: Solve the 2 variable system
(10x - z = 8) 29 -----------------> 290x -29z = 232
(34x - 29z = -24) -1 ------------> -34x + 29z = 24
+------------------256x = 256 -----> x = 1
Step 3: Plug value x into the 2 system equation to find z
10(1) - z = 8
z=2
Step 4: Plug in both values back into the 3 system equation to find y
7(1) + 5y - 3(2) = 16
5y = 15
y=3
Step 5: Give values of x, y, and z
{1,3,2}
Real World Applications:
The movie theater sold a total of 8500 movie tickets. Their earnings were $64,600. Tickets
can be bought in 3 ways: matinee admission costs $5, student admission is $6 all day, and
regular admissions are $8.50. How many of each type of ticket was sold if twice as many
student tickets were sold as matinee tickets?
Step 1: Define your variables
m = number of matinee tickets sold
s = number of student tickets sold
r = number of regular tickets sold
Step 2: Create your equations
r + s + m = 8500
8.5r + 6s + m = 64600
s = 2m
Step 3: Solve for a 2 variable system
(r + s + m = 8500) -6 ----> -6r - 6s - 6m = -51000
8.5 + 6s + 5m = 64600 ---> 8.5r + 6s + 5m = 64600
+------------------------2.5r - m = 13600
r + s + m = 8500
-s + 2m = 0
+------------------r + 3m = 8500
Step 4: Solve the 2 variable system:
(2.5r - m = 13600) 3 ------> 7.5r - 3m = 40800
r + 3m = 8500 --------------> r + 3m = 8500
+---------------------8.5r = 49300
r = 5800
Step 5: Plug ‘r’ into the 2 variable equation
5800 + 3m = 8500
3m = 2700
m = 900
Step 6: Plug in ‘r’ and ‘m’ into the 3 variable equation
5800 + s + 900 = 8500
s = 1800
Step 7: Write a sentence explaining your answer
The amount of tickets sold were 900 for the matinee, 1800 student tickets, and 5800 regular
tickets.
5-7 Quadratic & Linear Functions as Mathematical Models
Goal: find a mathematical model, quadratic or linear, for a real world situation
Find the particular equation of a quadratic equation from points on a graph to solve real
world problems.
The Krabby Patty has 3 prices for drinks.
Thimble (2oz) 1.35
Sip (8oz) 3.50
Taste (24oz) 4.25
a. Assume that the price is a quadratic function of the number of ounces. Write the
particular equation expressing price in terms of ounces.
b. If the Krabby Patty made a drink called Small that was 32 oz what would you predict
the price would be? Does this answer make sense?
c. Suppose they wanted another drink that cost 3.55. How many ounces would it be?
d. In what domain do you think this quadratic function will give reasonable values for
ounces?
e. What does c represent in this equation?
a. Let: o=the number of ounces
p=the price ​
in cents
2​
General equation: p=ax​
+bx+c
​
Ordered pairs: (2, 135), (8, 350), (24, 425)
Substituting the three ordered pairs:
4a+2b+c=135
64a+8b+c=350
576a+24b+c=425
Solving this equation we get:
a=-1.4157
b=49.9905 c=40.6818
2​
-1.4157o​
+49.9905o+40.6818=p
b. substitute 32 in for ounces
we get 190.701 which means that it would cost 1.91
No because it is cheaper than the drinks smaller than it.
c. we put 355 in for p and solve to get 27 ounces
d. x<18
e. the price of the cup and service charge, does not include price of liquid
Practice Problems:
1. When a volleyball is served, it goes up into the air, reaches a maximum height, then
goes back down. This is a quadratic function.
a. let t=number of seconds
b. let d= distance above the ground
c. The ball was served at 8 feet above the ground. One second later the ball is 14
feet above the ground. 2 seconds later the ball is 10 feet up. What is the
equation?
d. what does the vertex represent?
e. A volleyball net is 7 feet 4 inches. It is 1.5 seconds away from the endline. Will
the ball go over the net?
f. The length of a volleyball is 59 feet long. It takes 2.4 seconds for the ball to get
there Assume that the volleyball was served at the endline exactly. Will the
ball land in? (lines are in)
6-4 Exponentiation for Rational Exponents
Goal: Learn the definitions of exponentiation for rational exponents by evaluating powers and
by simplifying expressions involving rational expressions.
Exponentiation for Negative Exponents: ​
the
xa
xb
If
and a = b, then
xa
xb
expression x−n =
1
xn
= 0.
Exponentiation for 0 Exponent: ​
x⁰=
1, provided x =/ 0
Exponentiation for Reciprocal Exponents:
1
xn
= √n x
Definitions of the expression √n x :
n - root index
x - radicand
√ - radical sign
Exponentiation for Fractional Exponents (where a
​​
and ​
b​
are integers):
x b = ( √x a ) a
b
Sample Problems:
90x 19y 27
​ith positive exponents.
w
270x 33y 23
73
2. Simplify​
: 7 0 1. Simplify​
:
3. Express
4
√z 3 with a fractional exponent.
4. Am I. Wright attempted to simplify the product of
27x 39y 26
3x 3y 13
13 2
= 24x 1 y 27x 39y 26
3x 3y 13
:
= 24x13y 2
Is her answer correct? If not, show the steps to how you got the correct solution.
Answers:
1-3:
1. 12xy + 9y²
2-2:
1. a.
b.
4-6:
1. x = 1, y = 3, z = 2
2. m = 900, s = 1800, r = 5800
5-7:
2​
c. -5x​
+11x+8
d. the peak of the serve
e. yes
f. no
6-4:
1.
2.
3.
90x 19y 27
(2•3 2•5)y 4
33 23 =
(2•3 3•5)x 14
270x y 3
3
70 = 1. 7 7 0 = 71 = 73
4x 4
3x
3
√z 3x = z 4x 4 = z4x 3 4. Her answer is incorrect.
correct solution:
27x 39y 26
3x 3y 13
36 13
= 9x 1y = 9x36y 13
4
y = 3x 14