# Precalculus Course Objectives

**Course Goal:** The goal of Math 151: Precalculus is
to prepare each student for

Calculus Analytic Geometry I. A student well prepared for calculus will be able
to model

real-world problems and analyze their quantitative aspects using strong
algebraic and

trigonometric skills.

**Course Objectives:
**

**Throughout the semester, the student will:**

1: Identify various functions, including square root , cubic, and piecewise-defined

functions, and perform transformations and operations on known functions to create new

functions. Construct mathematical models from real-world word problems.

2: Identify quadratic, polynomial, and rational functions and analyze their graphs. Solve

polynomial and rational inequalities. Evaluate real-world problems using rational

function models.

3: Find the real zeros of polynomial functions using various theorems.

4: Identify one-to-one functions and find their inverses. Explore exponential and

logarithmic functions and use their properties to model and solve real-world problems

involving compound interest and uninhibited growth/decay.

5: Explore angles and their measure. Establish the value of trigonometric functions using

right triangle trigonometry. Analyze properties of the graphs of trigonometric functions

and apply transformations and shifts to sinusoidal graphs.

6: Explore inverse trigonometric functions. Prove identities involving trigonometric

functions and use them to solve trigonometric equations.

7: Solve real-world applications by creating right triangles and using trigonometric and

inverse trigonometric functions.

**Student Learning Outcomes by chapter:**

**Test questions listed will assess the student’s achievement of the desired
learning
outcomes. On completion of Math 151, the student should possess the ability to:**

**1:**

• Construct the graph of new functions by translating old ones (shifting,

stretching, reflecting)

• Construct the graph of new functions by translating old ones (shifting,

stretching, reflecting)

Ex. Write a function from its graph using transformations (shifts).

**• Analyze real-world situation by creating an equation
for a piecewise -defined
function and graph it**

Ex. At Car Rental R Us the following rate schedule applies for weekly rental of a

car.

Weekly rental fee | $135.00 |

Per mile charge | |

0 to 250 miles | $0.49/mile |

Over 250 miles | $0.29/mile |

(a) What is the charge (rounded to the nearest cent) for a weekly car

rental if the car was driven 175 miles?

(b) What is the charge (rounded to the nearest cent) for a weekly car rental if
the car

was driven 475 miles?

(c) Construct a piece-wise function that relates the weekly charge C for x miles

driven.

(d) Graph the function, label and scale the axes and several points on your
graph.

**• Formulate a composite function and find its domain**

Ex. Given the functions , and
find and simplify

.

**2:
• Use the maximum/minimum value of a quadratic function to optimize applied
problems**

Ex. A landscape engineer has 200 feet of border to enclose a rectangular pond.

What dimensions will result in the largest pond?

**• Evaluate applied problems using rational functions**

Ex. The cost function for manufacturing Flash Memory cards is estimated to be:

Where C(x) is the daily cost, and x is the number of flash
memory cards manufactured

per day.

Economists define the **average cost function** as

a) Find the average cost function **and** state the
domain.

b) What is the average cost (to the nearest penny) of manufacturing 50 flash
memory

cards per day?

c) How many flash memory cards would need to be printed per week if the average

cost per flash memory card is $20? Round your answer to the closest integer.

d) Find the number of flash memory cards manufactured per day that should be

produced to minimize the average cost. Round your answer to the closest integer.

e) What is the minimum average cost, to the nearest penny?

**• Solve a quadratic equation.**

Ex. Solve the equation. Find exact values where possible, otherwise round to two

decimal places .

**• Determine where a function is increasing/decreasing, and identify the
domain,
vertical/horizontal/oblique asymptotes of a rational function**

Ex. The concentration C (in milligrams) of a certain drug in a patient’s bloodstream

t minutes after injection is given by Using your graphing

calculator, graph C (t) to answer the following questions.

a.) What happens to the concentration of the drug as t increase?

b.) Determine the time at which the concentration is highest.

c.) What is the highest concentration?

d.) Determine the interval (s) where the function is increasing.

e.) Determine the interval (s) where the function is decreasing.

f.) What is the horizontal asymptote of C (t)?

If the drug is to be re-administered when the concentration decreases to 2 milligrams,

how many minutes after the initial dose is the drug re-administered?

• Solve polynomial and rational inequalities and graph them

• Solve polynomial and rational inequalities and graph them

Ex. Solve the inequality and graph the solution set on the real number line.

**3:
• Calculate the real zeros of a polynomial function by applying Descartes’ Rule
of
Signs and the Rational Zeros Theorem, and use the zeros and end behavior to
graph the polynomial**

Ex. Answer the following to analyze the graph of the polynomial function

a. What is the y- intercept ?

b. Find the exact zeros and complete the table for each zero of . ()

Zeros | Multiplicity |

c. What power function does f (x) resemble for large
values of l x l?

d. Using the information from a, b, and c), sketch the graph below. Label your
axes.

Label the points found in parts a and b.

**• Apply the remainder theorem and factor theorem to determine if a function
has
specified zeros**

Ex. Find k such that has the factor x-2

**4:**

• Calculate the time required to double or triple money or determine the future

value of a lump sum of money

• Calculate the time required to double or triple money or determine the future

value of a lump sum of money

Ex. How much will you have after 5 years if you invest $20,000 earning 4.75%

compounded a. Monthly and b. Continuously. Round your answers to the nearest

cent.

**• Develop equations of populations that obey the law of uninhibited growth or**

decay and predict future populations

decay and predict future populations

Ex. The half-life of a certain radioactive substance is about 2,750 years. If 50 grams is

present in a sample of soil now, how much will be present in 500 years? Round your

answer to the second decimal place.

**• Calculate the inverse of a function and construct its graph**

Ex. The function is one-to-one. Find the inverse of in explicit

form.

**• Construct exponential expressions from logarithmic
expressions and vice versa
to solve exponential and logarithmic equations. Using properties of logarithms
to
create a single logarithmic expression from a sum/difference of logarithms to
solve logarithmic equations.**

Ex.

**5:
• Acquire an angle measure in radians by converting from degrees to calculate
the arc length of a circle and the area of a sector of a circle**

Ex. The minute hand of a clock is 6 inches long. How far does the minute hand

move in 15 minutes? What is the area of the sector covered by the minute hand in

those 15 minutes?

**• Establish the remaining trigonometric functions given the value of one of them**

Ex. Find the exact value of the remaining trigonometric functions given that tan θ=3

and sin θ<0. Include a sketch to explain your work.

**• Acquire the value of trigonometric functions of acute angles (including 30, 45, 60**

degree angles) and general angles

degree angles) and general angles

Ex. Find the value of each remaining trig function of the acute angle given that

**• Apply the fundamental identities to simplify and
solve equations
**Ex. on the
interval

**• Apply the complementary angle theorem to acquire the
value of trig functions**

Ex. Find the exact value without using a calculator:

**• Apply coterminal angles to find the exact value of a
trig function**

Ex. Find the exact value without using a calculator:

**• Identify the reference angle and apply the theorem on
reference angles**

Ex. Find the exact value without using a calculator:
.

**• Determine the domain, range, amplitude, and period of
sinusoidal trigonometric
functions. Using given data, create a sinusoidal function.**

Ex. Write an equation in sine and cosine for the following graph. Two equations are

required.

**6:
• Calculate the exact value of the inverse sine, cosine, and tangent functions**

Ex. Find the exact value.

**• Verify identities**

Ex. Establish the identity:

**7:
• Solve right triangles**

Ex. Solve the right triangle.

Specify degrees or radians.

**• Assess applied problems using right triangles**

Ex. A laser beam is to be directed through a small hole in the center of

a circle of radius 10 feet. The origin of the beam is 35 feet from the circle
(see the figure). At

what angle of elevation should the beam be aimed to ensure that it goes through
the hole?

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