Pre Calculus: Week 13, Reciprocals of Linear Functions

This week in Pre calc 11 we learned how to graph Reciprocals of Linear Functions.

Reciprocal functions are graphed with  y= \frac{1}{x}

How to graph a Reciprocals of Linear Functions

Step 1:  Graph the original Linear function

Step 2:  Find the Invariant Points (The Invariant Points are where the line meets  y= -1 and y=1. )

Step 3: Find the Asymptotes and draw in dashed lines for both. There is a vertical and horizontal one. This is an imaginary line in which a graph reciprocal function will approach but will never reach. The vertical asymptote for the line will be in the middle of the two Invariant Points. The Invariant Points are where the line meets  y= -1 and y=1.

Step 4:  Draw the hyperbola.

Let’s take -2x +5 and \frac{1}{-2x + 5} for example

  • The horizontal asymptote is y=0
  • The vertical asymptotes is x = 2.5
    • A thing to notice when trying to find the vertical asymptote is that it is where the original line’s x axis is.
    • It is also in the middle of the two invariant points

The invariant points would be (2, 1) and (3, -1)

Things you will have to define for  \frac{1}{-2x + 5}

  • x intercept : none the reciprocated function does not cross the x axis because there is a horizontal asymptote
  • y axis: y= 0.2
  • Domain: XER,  x\neq 2.5
  • Range: YER, y\neq 0
  • Asymptotes:
    • horizontal : y = 0
    • vertical : x = 2.5

This is how you would graph Reciprocals of Linear Functions.

 

Conserving Water

Why is it important to conserve water in Canada?

It is important to save water in Canada because the majority of our land is farmland. We need to cut down on our water usage so we don’t run out of it and lower our other resources. Another reason why we should conserve water is so we don’t create more waste water and have to use more energy to clean it, as well as keeping our environment clean because our environment can only take so much and if we continue to pollute it then we are hurting the environment and hurting ourselves.

Something I already do to save water is turn the tap off when i brush my teeth. A new way for me to save water that i am going to implement into my life is take shorter showers as well as wash my face in the shower instead of doing it in the sink with the water running.

 

Pre Calculus Week 12:

This week in Pre-Calc 11 we started a new unit; Absolute Values and Reciprocal Functions. We continued off of the linear and quadratic functions we learned about, and how to graph these new kinds of functions. I will also show you how to right piecewise notation.

Reminder

linear function: y=mx+b

Quadratic: y= a(x-p)^2 + q

|Absolute value signs| : The distance away from zero, makes every number in between these lines positive.

When an absolute value symbol is added into an equation, it will force any part of the line or parabola that is in the bottom (negative) of the graph to flip and become positive. When this happens the point at where the line will have an immediate turning point also known as the point of inflection or as the critical point. This point usually has an x value and a y value of 0.

A few thing you will need to identify are:

  1. Point of Inflection (x intercept)
  2. y intercept
  3. Damain
  4. Range

Example: y= 4x +6 and y = |4x+6|

y=4x+6

Will turn into… when absolute value signs are added

Identify:

point of inflection:( -1.5, 0 )

y intercept: ( 0, 6 )

Domain: XER (because the graph goes on forever in each direction)

Range: y\underline{>} 0

Piecewise Notation

This a way to describe a function, and it has two parts.

  1. The first part, we write the original equation and when it is positive

 

 

 

 

 

This would be how to describe the positive part of the line

 

2.  The second part of the piecewise notation would be described by placing brackets around the equation and a negative in front to flip it to become positive 

The restriction helps to tell where the description occurs

3.  Now combine the two parts like…

This would be the complete way to write piecewise notation, and how to fully describe the function

Parabola

This would be very similar to a parabola

example: y = |1(x-5)^2 -1|

Would turn into…

 

Piecewise notation

The first part of the piecewise notation explains that any x value smaller than 4 will be positive and any number larger than 6 will also be positive.

The second part of the piecewise notation describes where the parabola would be negative, but it has been placed with the negative sign so it makes whatever is in the negative side positive. So any points between 4 and 6 would be negative, but the absolute value signs have made it.

positive.

 

This is how you would write piecewise notation as well as what a linear and quadratic function would look like on graphs when introduced with absolute value signs.

Poverty Life Cycles

POVERTY CYCLE OF HIV

  1. Baby Born with HIV
  2. Mother dies giving birth
  3. Child is unable to receive treatment
  4. Child cannot attend school because child has to work to get treatment 
  5. Child will start a family at a young age because no education

Solution: to have charities that help fund countries with medical care for free so families can save money and increase survival rates and allow kids to focus on education and be able to obtain a good job in the future.

WOMENS EQUALITY POVERY CYCLE

  1. Woman born into inequality
  2. Not treated fairly
  3. Woman gets taken advantage of at a young age (Rape) gets pregnant
  4. Woman cannot get education
  5. Woman only works at home

Solution: Allow girls to get education and make the education more generalized for everyone about poverty and health as well as normal school subjects. This will allow both genders to be aware of the possibilities of having children at an early age and will allow for woman to put off children until after education, which will allow for the woman to build a family without getting stuck in poverty.

CHILD BORN INTO POVERTY

  1. Child born into poverty
  2. Family cannot afford food
  3. Child needs to work for family
  4. Child receives no education
  5. Child has kids at a young age sue to lack of education/ family planning

Solutions: Make laws that protect child workers  and force companies to provide education for student workers for them to receive and education and help their development instead of damaging it.

Pre Calculus 11 Week 11: Graphing Linear Inequalities in Two Variables

This week in week 11 we started Graphing Inequalities. The majority of this is related to our graphing from last year in grade 10 math.

Linear graphs are known as straight line graphs with the equation of y=mx +b.

m is the slope ( \frac{rise}{run} )                                                                                                            b is the y intercept

For example: y =\frac{1}{2}x + 6 on a graph looks like:

Now when graphing inequalities you have to find a sign that will have a true statement after you have chosen a point on the graph to substitute x and y.

\star Note that > and < are broken lines on the graph and \underline{<} and \underline{>} are solid lines like the above graph ( because it is equal to so it includes the line)

  • 0 < \frac{1}{2}x + 6 (greater than zero)
  • 0 \underline{<} \frac{1}{2}x + 6 (greater or equal to zero)
  • 0 > \frac {1}{2}x + 6 (less than zero)
  • 0 \underline{>} \frac {1}{2}x + 6 (less than or equal to zero)

Now to graph the inequality you have to choose a point on the graph that makes the expression true for example:

y < \frac{1}{2}x + 6

\star to make things easy use (0,0) as the point

  •  y < \frac{1}{2}x + 6
  •  0 < \frac{1}{2}(0) + 6
  • 0 < 0 + 6
  • 0 < 6

This statement is true because 0 is smaller than 6. This means that the side that has the coordinate of (0,0) will be shaded in. This will also have a broken line because it is not equal to.

Now when we change the sign to greater to (>) it will flip because 0 would not be greater than 6

 

  • y > \frac{1}{2}x + 6
  •  0 > \frac{1}{2}(0) + 6
  • 0 > 0 + 6
  • 0 > 6
  • FALSE STATEMENT

If the sign was \underline {>} or \underline {<} the line would be solid.

You can also graph a parabola

example: y = x^2 + 4x + 2

Step one: Graph the parabola by putting it into standard form.

  • y = x^2 + 4x + 2
  • y =\underbrace{x^2 + 4x + 4} - 4 + 2
  • y = (x + 2)^2 -4 + 2
  • y = (x + 2)^2 -2

Graph it from here.

Step two: chose a point on the graph and make a true statement.

  • y \underline{<} x^2+4x+2
  • 0 \underline{<} 0 + 0 +2
  • 0 \underline{<} 2
  • TRUE STATEMENT, 0 is smaller than two

The side with (0,0) as a coordinate will be shaded in and the line will be solid because it is equal to.

Now if the sign was change the inside of the parabola would be shaded in.

This is how to graph inequalities with linear equations as well as quadratic equations.

Pre Calculus 11 Week 10: Reviewing Factoring with Substitution

Last week in Pre Calculus 11 we mostly reviewed for our midterm exam. As I was studying I came across a few things that I totally forgot about. Things that would help make my math life easier, like how to use substitution when factoring. This is helpful because it is simple, faster, and not as messy as factoring without this trick. I will quickly review how to do this.

Example: 3(2x-1)^2 + 14(2x-1) + 8

Step 1: Substitute (2x-1) with a variable.

\star Notice that there are two (2x-1), therefore you can use the same variable for both terms.

  • 3a^2 + 14a + 8

Step 2: Factor3a^2 + 14a + 8

  • (3a + 2) (a + 4)

Step 3: Substitute (2x – 1) back into the factored form to replace the variable a.

  • (3a + 2) (a + 4)
  • (3(2x -1) + 2) ((2x-1)+ 4

Step 4: Distribute

  • (3(2x -1) + 2) ((2x-1)+ 4
  • (6x-3 +2)(2x-1+4)

Step 5: Simplify

  • (6x-3 +2)(2x-1+4)
  • (6x-1)(2x+3)

FINAL ANSWER

(6x-1)(2x+3)

To find the solutions make each factor equal to zero and solve for x:

6x-1=0 –> 6x=1 –> x=\frac{1}{6}

2x + 3=0 –>2x=-3 –> x =- \frac{3}{2}

This is how to use substitution while factoring. This is a very helpful and quick way to factor difficult looking equations.

 

Week 9: How to convert from General form to Standard form

This week in Pre Calc 11 I learned how to convert general form into standard form. I will show you how it uses completing the square but it a little different from last unit of solving quadratic equations. In this unit it helps us graph what the function will look like (this should look like a parabola)

Converting from General Form to Standard form.

ex/ 2x^2 -11x +4

Step 1: Place brackets around 2x^2-11x , then divide by 2 to getx^2 instead of 2x^2

  • (2x^2 -11x) +4
  • 2(x^2 - \frac{11}{2}x ) +4

Step 2/3: Divide \frac{11}{2}x by 2 and square it; place it in two blank spaces beside \frac{11}{2}x one being added and one being subtracted.

  • 2(x^2 - \frac{11}{2}x +{blank} - {blank}) +4
  • 2(x^2 - \frac{11}{2}x +\frac{121}{16} - \frac{121}{16}) +4

Step 4: Multiply the 2 you factored out in step 1 to \frac{121}{16} to remove it from the brackets.

  • 2(x^2 - \frac{11}{2}x +\frac{121}{16}) +4 -\frac{242}{16}

Step 5: Factor the inside of the brackets.

\star Hint: What you squared in step 2/3 is you factor.

  • 2(x^2 - \frac{11}{2}x +\frac{121}{16}) +4 -2\frac{242}{16}
  • 2(x - \frac{11}{4})^2 -\frac{242}{16} + 4

Step 6: Simplify

  • 2(x - \frac{11}{4})^2 -\frac{242}{16} + 4
  • 2(x - \frac{11}{4})^2 -\frac{121}{8} + 4
  • 2(x - \frac{11}{4})^2 -\frac{121}{8} + \frac{32}{8}
  • 2(x - \frac{11}{4})^2 -\frac{89}{8}

Final Answer: 2(x - \frac{11}{4})^2 -\frac{89}{8}

You can check if you have done your calculations correctly by using desmos.com

\star notice how both equations line up perfectly, this means that both are equivalent.

Standard form can tell us:

Standard formula: y= a(x-p)^2 +q

a = 2

p\frac{11}{4}

q-\frac{89}{8}

Horizontal translation: \frac{11}{4} units right

Vertical translation: -\frac{89}{8}

Vertex : (\frac{11}{4} , -\frac{89}{8})

Axis of symmetry\frac{11}{4}

This is how you convert General form into standard form in the unit of Quadratic equations. The standard form can tell you a lot about what it looks like and how to graph it. This is my favorite form in this unit because it tells me so much information and it is very useful. 🙂

 

5 year Age Groups and Gender of Benin

  1. Find the dependency ratio, is it high or low?
    • 83.5% dependent. It is high.
  2. Describe the situation in the country based on the info in your population pyramid (births, deaths, health, age, male/female, type of pyramid, stage in DTM etc.)
    • High births, which are going to be dependent on the working age cohort, we have more kids than adults. We have less deaths than births; meaning they are getting an enhanced health care system/ everything is evolving. There are slightly more women than men in this country. This is an expanding pyramid. This is around stage 2.
  3. Explain what the country needs to prepare for in the near future and why you think that. (health, population, business, policies, etc. Connect this to your observations)
    • This country needs to keep making medicine, needs to increase food production, needs to make more schools and hospitals. They might need to make a policy of how many kids you are allowed to have if the population continues to increase at the rate its going. The health of these people are generally good but they die off around 80, but since there are high birth rates the death rate will increase later on so the country might want to make room for cemeteries and they might want to start businesses that help elderly people, like transportation or have more senior homes.

Week 8: General Form, Quadratics

This week I learned many cool and fascinating facts about the general form of a quadratic equation ( ax^2 + bx +c ) as well as analyzing y=a (x-p)^2 + q

General form

x^2 :

  • Positive – Open upward
  • Negative – Opens downwards

bx:

  • is a mixture of both ax^2 and bx so it can differ

c:

  • Is the y axis

Analyzing equation:  y=a (x-p)^2 + q

using this formula can tell us a lot of things that will help us graph the equation. In fact it will tell us 8 helpful clues.

ex. $latex -2(x+2)^2 – 1

  1. Vertex: the vertex will be p and q … (-2,-1) * p is always backwards from the formula because in the original equation it is (x – p) therefore it has to be a negative number for it to become positive in the new equation.
  2. Axis of symmetry: This will be p because it is the x value on a graph …-2
  3. Opens up or down: This is determined if x^2 is negative or positive. If it is negative it will be opening down, and if it is positive it will be opening upwards… Opens down because it is -2
  4. If it is congruent to y = x^2. This means that the pattern of the graph would be 1, 3, 5. Meaning 1 over 1 up, 1 over 3 up, 1 over 5 up etc. If it does not follow the 1,3,5 rule it means that a has a value different from -1 or 1. Say a = 4, multiply 4 to each number from the original pattern. ex/ 1,3,5 is now equal to 4, 12 , 20 etc. This pattern will also tell us if the parabola will be stretched or compressed. If 1> a < 0 ( fraction ) it means that the parabola will be compressed. If  1 < a > 1 it means that the parabola will be stretched…. this equation is congruent to y = -2x^2, it is being stretched because 2 > 1, it is negative so it will be flipped upside down but the structure is the same. (congruent)
  5. Minimum or maximum: If a is positive, it will have a minimum. If a is negative it will have a maximum… It has a maximum of -1
  6. Domain. We know the XER because the parabola never ends.
  7. Range: If a is positive y will be greater than the minimum. If a is negative, y will be smaller than the maximum… y < -1, yER

IMPORTANT

$latex y = 3x^2 (x – 3)^2 + 4

The horizontal translation will be 3 units to the right even though it says -3.

 

 

I also learned that general form can be turned into the analyzing equation of  y=a (x-p)^2 + q by completing the square.

Here is a video that helped me understand how to complete the square

=

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