Down to Two Classes

Week 8:

Math 108

Reasoning Process help solve many different types of problems encountered in every day life such as:

How long will it take me to pay off this loan?
How much money should I save each month if I want to buy a car?
How much money should I save each month in order to retire at age 62?
How many calories should I eat each day if I want to compete in a triathlon?
Should I buy a car with better gas mileage?
Should I accept a job offer that pays commission rather than a salary?
Which health insurance plan is best for my situation?
How much money do I need to start a community charity?

Linear functions help us describe or mathematically model real world situations where something is changing at a constant rate.

A linear function is a function of the form $f (x) = m x + b$ where $m$ and $b$ are constants.

Two features that help us graph a linear function are the slope and the y-intercept. The equation $f (x) = m x + b$ is said to be in slope-intercept form.

The $y$ -intercept is the point on the graph where the line crosses the vertical axis ( $y$ -axis).
The slope of a line is a number that tells us how steep the line is. If the slope is $\frac{2}{3}$ , starting from any point on the line, we go up 2 units for every 3 units we go to the right. If the slope is $- 4$ , we would start at any point on the line and then go down 4 units for each 1 unit we go to the right.

The graph of a linear function,

f (x) = m x + b

is a straight line that crosses the

y

-axis at the point

(0, b)

and has a slope of m.

m

The function $3 x + 4 y = - 7$ is also a linear function. It is not currently in the form $f (x) = m x + b$ , but we could rewrite it in this form using some algebra rules. First, we solve the equation for $y$ :

A Linear Application - Calories per Day

A calorie is a unit of measurement that we use to measure energy. Sarah is a 22-year-old woman with an average weight and height but who does very little physical activity. Nevertheless, in a given day, she spends about 1860 calories of energy.2 This means if Sarah eats more than 1860 calories in a day, her body will store the extra calories and she will gain weight. If she eats less than 1860 calories in a day, her body will need to use some of the stored calories to make up the difference and she will lose weight.

Sarah is trying to improve her health and realizes that by exercising, she can increase her caloric expenditure. Sarah decides to start walking each morning before she goes to work. She has determined that for her weight and the speed (15 minutes per mile), she will burn about 6 calories for each minute she walks3.

Her schedule varies every day so she can’t always walk for the same amount of time. However, Sarah realizes that she can create a linear function that will give her the total amount of calories she can consume that is based on the number of minutes she walks. This will allow her to better monitor her caloric expenditure and meet her health goals.

The number of calories she spends on a day she doesn’t go for a walk, 1860 calories, will be the $y$ -intercept of the linear function. The slope of the function will be the number of calories burned per minute, or 6 calories per minute. This gives her the following linear function:

Sarah makes the following chart to determine how many total calories she would need each day, depending on how long her walk is:

$x$ = Length of Walk	$f (x)$ = Total calories
5 minutes	1890 calories
10 minutes	1920 calories
15 minutes	1950 calories
20 minutes	1980 calories
25 minutes	2010 calories
30 minutes	2040 calories
35 minutes	2070 calories
40 minutes	2100 calories
45 minutes	2130 calories
50 minutes	2160 calories
55 minutes	2190 calories
60 minutes	2220 calories

On a day when Sarah walked 30 minutes prior to work, she used the following graph to help her see how the walk effected the total number of calories she would burn that day. Based on this graph, she knows that her caloric expenditure is 2040 calories.

A quadratic function is a function in the form $f (x) = a x^{2} + b x + c$ where $a, b$ , and $c$ are constants.

The graph of a quadratic function is always a $u$ -shaped curve called a parabola. Three features that help us graph a quadratic function are the $y$ -intercept, the direction the parabola opens, and the vertex of the parabola.

The y-intercept is the point on the graph where the parabola crosses the vertical axis ( $y$
The direction of the parabola refers to whether the u-shape of the parabola opens up or opens down.
The vertex of the parabola is either the highest or the lowest point on the parabola (depending on the direction of the parabola).

The graph of a quadratic function is a parabola. The vertex of the parabola has an $x$ -coordinate of $x = \frac{- b}{2 a}$ . You find the $y$ -coordinate of the vertex by finding $f (\frac{- b}{2 a})$ . The y-intercept is at $(0, c)$ . If $a$ is positive, the parabola opens up. If $a$ is negative, the parabola opens down.

Example 9

$x^{2} - 2 y = 3 x$

Answer:

Both equations a. and c. are quadratic functions.

Notice that equation a. is already written in the form $f (x) = a x^{2} + b x + c$ . In this case $a = - 4$ , $b = 2$ , and $c = - 3$ .

In equation c. the function is not written in standard form, but it can be rewritten as $f (x) = \frac{1}{2} x^{2} - \frac{3}{2} x + 0$

Exponential functions are used in many real world applications such as computing compound interest in bank accounts and modeling the growth of populations.

An Exponential Function is a function in the form $f (x) = a \cdot b^{x} + c$ where $a, b$ , and $c$ are constants and $b > 0$ and $b \neq 1$ .

There are three features that help graph exponential functions:

The y-intercept is the point where the graph crosses the vertical axis ( $y$ -axis).
The horizontal asymptote is a line that the function gets closer and closer to, but never crosses.
The graph can be either increasing or decreasing. An increasing graph goes up as you move along the graph from left to right. A decreasing graph goes down as you move along the graph from left to right.

The graph of an exponential function has a

y

-intercept at

(0, a + c)

. Plotting another point on the graph helps determine the steepness of the graph. The graph has a horizontal asymptote at

y = c

. When

0 < b < 1

the graph is decreasing when

b > 1

it is increasing.

Jesus Christ and the Everlasting Gospel

This week you will gain a greater understanding of God's great plan for our lives by studying the following truths:

The only way we can return to live with Heavenly Father is by following Jesus Christ.
Jesus Christ invites all people to come unto Him and to be His disciples.
Discipleship requires our sustained willingness to forsake all and follow Jesus Christ.
Jesus Christ was persecuted for doing good.
As we seek to follow the Savior’s example of doing good, we will sometimes have to endure persecution.
If we are to follow Jesus Christ’s teachings, we must learn to love our enemies and be kind to those who persecute us.
As we follow the Savior’s example, we will be blessed with courage to live and defend our faith, and we will be able to help others draw closer to the Lord.
During His mortal ministry, the Savior called, ordained, and commissioned twelve apostles.
Jesus gave His Apostles priesthood authority that enabled them to do the same works they had seen Him do.
Jesus conferred priesthood keys upon His Apostles so they would have authority to direct the Church both before and after His death.
Priesthood keys are critical in establishing and maintaining order in the Lord’s Church, thus helping to bring to pass the immortality and eternal life of man.

The word disciple and the word discipline share a common Latin root—discipulus—which means pupil. In the gospel of Jesus Christ, a disciple is defined as someone who is a pupil of the Savior Jesus Christ. Disciples of the Savior “receiveth [His] law and doeth it” (D&C 41:5 (Links to an external site.)). Disciples also seem interested in sharing their knowledge and experience with others. One example of this is when Mormon declares, “I am a disciple of Jesus Christ, the Son of God, I have been called of him to declare his word among his people, that they might have everlasting life” (3 Nephi 5:13 (Links to an external site.)). The Savior is the perfect example of one who came into the world to do the will of the Father and actually did and taught the will of the Father in perfection or fullness.

This week provides opportunities to become more familiar with the Savior’s teachings about and examples of discipleship. One thing true disciples of the Savior come to recognize about discipleship is the need to find the Lord’s authorized servants to guide them in the ways of truth and salvation. Joseph Smith taught, “Whenever men can find out the will of God and find an administrator legally authorized from God, there is the kingdom of God; but where these are not, the kingdom of God is not…for nothing will save a man but a legal administrator” (HC 5:256–59 (Links to an external site.)). Understanding true discipleship is a quest to understanding the role of legally authorized administrators of God and the blessing of finding and following someone who holds priesthood keys.

Instructions

Study the following scriptures and resources in preparation for this week's activities. Here are a few questions to ponder as you study:

What does it mean to be a disciple of Jesus Christ?

A disciple is more than just a follower of Christ. A disciple teaches about Christ in their words, thoughts, and actions. A disciple leads the way Christ wants them to lead, not for their own personal glory. A disciple of Jesus Christ knows that they are imperfect and fully relies on the atonement of the Savior. In order to be a proper or full time disciple, you need to be baptized and confirmed by the right authority. Then you need to be worthy to receive the priesthood or power of God, again by the right authority.

When was a time when you were prompted to increase your discipleship, and what changes did you make to follow that prompting?

As a young adult I had the opportunity to participate in Personal Progress. Growing up I didn't really know how to properly handle and release my anger. I had little patience and contention with my siblings was high. I loved Personal Progress but kept avoiding projects that seemed too hared. I decided to take baby steps when it came to being a peacemaker. I was in a habit of responding poorly when irritated or offended so I decided to count to 4, if by then I had cooled down enough to respond in a rational manner then I would. But if I felt that my response would only cause contention or make matters worse then I would leave. I still have a lot to learn in my path to discipleship but I am forever grateful to Personal Progress for helping me control my temper.

What is the difference between knowing about discipleship and the gospel and becoming a disciple?

My uncle had a strong knowledge of the gospel when he served his mission just out of high school. He was very influencial when he bore his testimony and converted many people to the church. My uncle has been inactive for many, many years now and when my Grandmother asked him how he was able to bear such a strong testimony, he said that he was just leaning on his parents testimony and telling people what he thought they wanted to hear. There is a big difference between knowing the gospel and becoming a disciple. Knowledge only becomes power when it is rightly applied. Matthew 7: 21-23 explains the difference between knowing and doing really well.

HOW CAN I NOT TAKE IT SO PERSONALLY WHEN PEOPLE REJECT THE GOSPEL OR REFUSE TO OBEY SPECIFIC COMMANDMENTS?

DO MEN AND WOMEN HAVE THE SAME SPIRITUAL POWER, EVEN THOUGH WOMEN DON'T HOLD THE PRIESTHOOD?

Study Material:

Matthew 5–7 (Links to an external site.) (3 Nephi 12:1–12 (Links to an external site.))

Matthew 22:34–40 (Links to an external site.)

Luke 10:25–37 (Links to an external site.)

Dallin H. Oaks, "The Challenge to Become," (Links to an external site.) Ensign, November 2000

Matthew 10:1–8

Matthew 16:15–19

Matthew 17:1–8

Jeffrey R. Holland, "Prophets, Seers, and Revelators," Ensign, November 2004