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P.4 Lines in the Plane
PreCalculus
P.4 LINES IN THE PLANE
Learning Targets for P.4
1. Calculate Average Rate of Change between 2 points
2. Write the equation of a line in any form (point-slope, slope intercept or general)
3. Understand how the slopes of parallel lines are related
4. Understand how the slopes of perpendicular lines are related
Slope Information to Know
The slope of a non-vertical line is given by
y y2  y1
…also known as average rate of change.

x x2  x1
•
•
A vertical line has undefined slope.
A horizontal line has slope of zero.
•
•
Parallel Lines have equal slopes.
Perpendicular Lines have slopes that are opposite reciprocals.
Equations of a Line (3 forms): To write an equation of a line, all you need is a POINT and the SLOPE of the line.
Point – Slope Form: y − y1 = m( x − x1 ) tells the slope and a point.
THIS IS THE EASIEST EQUATION TO WRITE…ALWAYS USE IT FIRST!!
Slope – Intercept Form: =
y mx + b tells the slope and y-intercept.
General (or Standard) Form: Ax + By =
C or Ax + By + C =
0 where all coefficients and constants are integers
Basically, general form has NO decimals, NO fractions and the x & y terms are on the same side of the equation.
And now to practice these concepts…
1. Suppose a line has a slope of  23 and passes through the point (–4, 3).
a) Write the equation of the line in point-slope form.
b) Write your answer from part a so that y is a function of x.
c) Change your equation into slope-intercept form by using the distributive property and combining like terms.
d) Explain why this equation is NOT in general form, and then “fix” it so that the equation is in general form.
Unit 3 - 1
P.4 Lines in the Plane
PreCalculus
2. Find the general form of the line passing through (1, 3) and having the following characteristics:
a) Parallel to the line 5x – 2y = 0
b) Perpendicular to the line 5x – 2y = 0
3. The velocity of a rocket launched when t = 0 was recorded for selected values of t over the interval 0 < t < 80.
Acceleration is the rate of change in velocity (a.k.a. slope of velocity). Finding average acceleration between two points
can be done by finding the slope between those two points on the velocity function.
t (seconds)
Velocity (feet/second)
0
5
10
14
20
22
30
29
40
35
50
40
60
44
70
47
80
49
Find the average acceleration of the rocket from t = 0 to t = 80. What are the units of measurement?
4. On a recent trip, you noticed that after 2 hours you were 90 miles from home, but after 5 hours you were 270 miles from
home.
a) Find the average rate of change from your first to your second observation.
b) What are the units of measurement? What does the average rate of change in your position tell you?
5. For the table in question 3, use a graphing calculator to find the linear regression equation for the velocity of the rocket
as a function of the time after launch.
Unit 3 - 2
1.6 Modeling with Functions
PreCalculus
1.6 MODELING WITH FUNCTIONS
Lesson Targets for 1.6
1. Change English statements into mathematical expressions
2. Write equations to model given situations.
3. Use equations to solve percentage and mixture problems.
Example 1: Write a mathematical expression for the quantity described verbally. (An expression has no equal sign, and
can therefore NOT be solved.)
a) A number x decreased by six and
then doubled.
b) A salary after a 4.4% increase, if the original
salary is x dollars.
Example 2: The diameter of a right circular cylinder equals half its height. Write the volume of the cylinder as a function
of its height. The volume of a right circular cylinder is given by V   r 2 h .
Example 3: For each statement below, do the following:
1. Write an equation (be sure to define any variables used).
2. Solve the equation, and answer the question.
a) One positive number is twice another positive number. The sum of the two numbers is 390. Find the two
numbers.
b) Joe Pearlman received a 3.5% pay decrease. His salary after the decrease was $27,985. What was his salary
before the decrease?
c) Investment returns: Jackie invests $25,000 part at 5.5% annual interest and the balance at 8.3% annual interest.
How much is invested at each rate if Jackie receives a 1-year interest payment of $1571?
Unit 3 - 3
2.2 Power Functions with Modeling
PreCalculus
2.2 POWER FUNCTIONS WITH MODELING
Learning Targets:
1. Identify a power functions.
2. Model power functions using the regression capabilities of your calculator.
3. Understand the difference between “direct variation” and “inverse variation”.
4. Use power functions to solve word problems.
In this section we will study power functions.
A power function looks like ____________________,
where a = _______________________________________, and k = _________________________________________.
We say that f (x) ______________ as the ath power of x, or that f (x) is ___________________________ the ath power of x.
If a is ______________________ we say that f (x) varies ______________________ with the ath power of x.
If a is ______________________ we say that f (x) varies ______________________ with the ath power of x.
Example 1: Determine if the function is a power function. For those that are not, explain why not.
a) f ( x ) = −3 x 4
b) f ( x ) = 3 8 x 5
c) g ( x )= 7 ⋅ 2 x
d) h ( x ) = 2 x −5
Example 2: The volume V of a sphere varies directly as the cube of the radius r. When the radius of a sphere is 6 cm, the
volume is 904.779 cm3. What is the radius of a sphere whose volume is 268.083 cm3?
Example 3: The force of gravity F acting on an object is inversely proportional to the square of the distance d from the
object to the center of the earth. Write an equation that models this situation.
Example 4: Velma and Reggie gathered the data in the table below using a 100-watt light bulb and a Calculator-Based
Laboratory(CBL) with a light-intensity probe.
a) Use your calculator to find the power regression model of the data.
b) Describe the relationship between the intensity and distance
modeled with the equation in part a.
c) Use the regression model from part a to predict the intensity of an
object 2.75 meters away.
Unit 3 - 4
Light Intensity Data for a 100-W
Light Bulb
Distance(m)
Intensity(W/m²)
1.0
7.95
1.5
3.53
2.0
2.01
2.5
1.27
3.0
0.90
2.1 Linear and Quadratic Functions and Modeling
PreCalculus
2.1 LINEAR AND QUADRATIC FUNCTIONS AND MODELING
Learning Targets:
1. Find the vertex and graph a quadratic function in standard, intercept, and vertex forms.
2. Model Quadratic functions in vertex form.
3. Use the projectile motion model to find the highest point a projectile reaches, and when it reaches that point.
Example 2: The table below shows the relationship between the number of Calories and the number of grams of fat in 9
different hamburgers from various fast-food restaurants.
Calories
Fat (g)
720
46
530
30
510
27
500
26
305
13
410
20
440
25
320
13
598
26
a) Find the linear regression model for this data.
b) According to your model, how much fat is in a hamburger with 475 calories? Do NOT round!!
c) In the context of this problem, what does the slope mean?
Example 3: Find the vertex of the quadratic function f ( x ) = 3 ( x + 5 ) − 2
Then, graph the quadratic function.
2
y
x
Many of us know to use x   2ba to find the vertex when in standard form … y  ax 2  bx  c .
Let’s see why this works…
Example 4: Start with the function f  x   a  x  h   k .
2
a) Multiply to expand  x  h  .
2
b) Distribute the a
c) Compare the coefficients of x 2 and x to a and b from the standard form equation.
Unit 3 - 5
2.1 Linear and Quadratic Functions and Modeling
PreCalculus
Example 5: Find the vertex of the quadratic function f ( x ) = 2 x 2 − 6 x + 11
y
and then graph it.
x
Modeling Quadratic Functions
•
•
In previous courses, we have used quadratic functions to model projectile motion. We will continue to use
quadratics in this manner through applications of the vertex and the zeros.
There are many other situations where we use quadratic functions to model real world applications. One such
application is shown below.
Example 6: The Welcome Home apartment rental company has 1600 units available, of which 800 are currently rented at
$300 per month. A market survey indicates that each $5 decrease in monthly rent will result in 20 new leases.
a) Determine a polynomial function R (x) that models the total rental income realized by Welcome Home, where x
is the number of $5 decreases in monthly rent.
b) Find a graph of R (x) for the rent levels between $175 and $300 (that is 0 < x < 25) that clearly shows a
maximum for R (x). What rent will yield Welcome Home the maximum monthly income?
Unit 3 - 6
3.2 Exponential and Logistic Modeling
PreCalculus
3.2 EXPONENTIAL AND LOGISTIC MODELING and USING THE FINANCE APP
Learning Targets:
1.
2.
3.
4.
Write an exponential growth or decay model f ( x)  ab and use it to answer questions in context.
Recognize a logistic growth function and when it is appropriate to use.
Use a logistic growth model to answer questions in context.
Write a logistic growth function given the y-intercept, both horizontal asymptotes, and another point.
x
Example 1: The half-life of a certain radioactive substance is 14 days. There are 10 grams present initially.
a) Express the amount of substance remaining as an exponential function of time. Define your variables.
b) When will there be less than 1 gram remaining?
Do you think it is reasonable for a population to grow exponentially indefinitely?
Logistic Growth Functions … functions that model situations where exponential growth is limited.
An equation of the form __________________________ or _____________________________.
where c = ________________________________.
The graph of a logistic function looks like an exponential function at first, but then “levels off” at y = c. Remember from
our Parent Functions in Unit 2, that the logistic function has two HA: y = 0 and y = c.
Example 2: The number of students infected with flu after t days at Springfield High School is modeled by the following
function:
1600
P t  
1  99e0.4t
a) What was the initial number of infected students t = 0?
b) After 5 days, how many students will be infected?
c) What is the maximum number of students that will be infected?
Unit 3 - 7
3.2 Exponential and Logistic Modeling
Example 3: Find a logistic equation of the form y
PreCalculus
c
1 ae
bx
that fits the graph below, if the y-intercept is 5 and the point
(24, 135) is on the curve.
The above problem can be done (sometimes without a calculator) using f ( x) 
c
1  ab x
instead of the continuous logistic
growth model …and we will practice this on the Worksheet.
Using the FINANCE App On Your TI-83 or TI-84 Calculator
Your graphing calculator has a very powerful financial calculator application built into its programming called the TVM
Solver. TVM stands for Time–Value–of–Money. The application solves for one variable when given values for any four of
the five remaining variables. It is especially useful when you are making the same periodic payment (or investment) more
than once a year…i.e. car payment.
First we need to understand the 5 variables involved…Press APPS, Finance, TVM Solver to get started…
1. n = the number of compounding periods
2. I% = the annual percentage rate
3. PV = the Present Value of the account or investment
4. PMT = the payment amount
5. FV = the Future Value of the account or investment
6. P/Y = payments per year
7. C/Y = compounding periods per year…for our use C/Y = P/Y
8. PMT: END BEGIN means payment at the end or beginning of the
payment period.
NOTE: Enter any money received in as positive numbers and any money paid out as negative numbers.
Example 4: Sal opened a retirement account and pays $50 at the end of each quarter (4 times/year) into it. The account
earns 6.12% annual interest. Sal plans to retire in 15 years. How much money will be in his account at that time?
Unit 3 - 8
3.5 Solving with Exponential, Logistic and Logarithmic Models
PreCalculus
3.5 SOLVING WITH EXPONENTIAL, LOGISTIC AND LOGARITHMIC MODELS
Learning Targets:
1. Solve exponential, logistic and logarithmic models and understand the context for these problems.
Recall from Unit 1 and Unit 2…exponents and logarithms are inverses of each other and we use them to solve equations
 
blogb M  M … and … log b b M  M
Example 1: The radioactive isotope Germanium-71 decays at a rate of 6% per day. A scientist has an initial
amount of 15 grams. How long will it take for half the same to remain? Solve algebraically.
Example 2: Often environmentalists will capture an endangered species and transport the species to a controlled
environment where the species can produce offspring and regenerate its population (ever watch Rio?). Suppose
that eight American bald eagles are captured, transported to Montana, and set free. Based on experience, the
environmentalists expect the population to grow according to the model below where t is measured in years.
How long does it take for the population to reach 250 bald eagles? Solve Algebraically
P t  
500
1  61.5e0.162t
Example 3: The Beer-Lambert law of absorption applied to Lake Erie states that the light intensity I (in lumens)
I 
at a depth of x feet, satisfies the equation log    0.00235 x . Find the intensity of the light at a depth of 30
12 
feet.
Unit 3 - 9
4.8 Sinusoidal Modeling
Pre-Calculus
4.8 SINUSOIDAL MODELING
Learning Targets
1. Model periodic phenomenon with an equation involving sine or cosine.
2. Use the model to make predictions about the real world.
3. Solve problems involving simple harmonic motion.
4. Model sinusoidal functions using the regression capabilities of your calculator.
You will recall the two basic trigonometric functions were f ( x)  sin x and f ( x)  cos x . Thinking of
transformations, the function f ( x)  cos x can be thought of as a sine curve with a phase shift of _____________.
We use the name ____________________________ for any function whose graph is a transformed a sine curve
(including the graph of cosine).
Example 1: The Ferris wheel shown makes one complete
turn every 20 seconds. A rider’s height, h, above the ground
can be modeled by the equation h a sin t k , where h
and k are given in feet and t is in seconds.
25 feet
a) What is the value of a ?
b) What is the value of k ?
c) What is the value of
8 feet
?
d) Write the model for the rider’s height as a function of the time.
e) Using this model, where is the person when they first get on the ride? Is this realistic?
f) Write a different equation for h that might be a little more realistic in terms of where a person gets on the
ride.
4-18
4.8 Sinusoidal Modeling
Pre-Calculus
Example 2: Naturalists find that populations of some types of predatory animals vary periodically with time.
Assume a sinusoid models the population of foxes in a certain forest as a function of time after the records started
being kept (t = 0 years). A minimum of 200 foxes appeared when t = 2.9 years. The next maximum, 800 foxes,
occurred at t = 5.1 years.
a) Sketch a graph of the sinusoid that
would model the fox population.
b) Determine an equation modeling
the number of foxes as a function of time.
c) Predict the fox population when t = 7 and 10 years.
Sinusoidal Regression
Example 3: The average discharge rate of the Alabama River is given in the
table for the odd-numbered months of the year.
a) Find a sinusoidal regression model and sketch it over the
given scatter plot below. The window shows 0  x  12
and 0  f ( x)  2000 .
Month (Jan = 1)
1
3
5
7
9
11
Rate (m3/sec)
1569
1781
1333
401
261
678
b) Assuming the model continues for the next calendar year, estimate the flow rate for the
following June.
c) Use the graph and equation to estimate the number of days per year the flow rate is below
500 m3/sec.
4-19