Quadratic Functions: Finding the Vertex from Vertex Form

Tutoring math 11 – which is needed for nursing, among other careers – you’ll need to explain how to identify the vertex of a quadratic function.

Vertex form is designed to easily yield the vertex of a quadratic function.  A quadratic function of the form

y=a(x-p) ² + q

has vertex at (p,q).

Example 1:  Find the vertex of y=-3(x-4)² +9

Solution: the vertex is at (4,9).

Notice the (“opposite, same”) pattern: the x-coordinate is opposite to what you see in the brackets, whereas the y-coordinate is the same as what you see added (or subtracted) at the end.

Example 2: Find the vertex of y=2(x+5)² -3

Solution: the vertex is at (-5,-3).

Notice that the number multiplying in front of the brackets does not affect the vertex.

Example 3: Find the vertex of y=(x-5)²

Solution: Remembering the form y=a(x-p)² +q, we need to discern the values of p and q. Clearly, p=5. q=0, because

y=(x-5)²

can also be written as

y=(x-5)² + 0.

Therefore, the vertex is at (5,0).

Example 4: Find the vertex of y=3x² + 7

Solution: Going back to y=a(x-p) ² + q, we realize that although q=7, we seem to be missing p. However, we can rewrite our equation as y=3(x-0)² + 7. Now, we realize that p=0. The vertex is at (0,7).

Identifying the vertex can be tricky, but I hope this helps.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC

Equation Solving: the legendary “cross-multiplying” method

As a math tutor, you’re bound to teach this quick method for solving proportions.

When you encounter

what do you do? Well, the easiest approach is probably what many call “cross multiplying”. I learned it in grade 7.

Cross multiplying argues that when two fractions are equivalent, their diagonals multiply to the the same amount. For instance, consider

Notice that the diagonals both mutliply to 10:

We can use the handy principle that equal fractions have equal diagonals to solve virtually any rate or proportion question – including percents.
Consider, again, our first problem:

We now know that, since the left fraction equals the right fraction, the diagonals must multiply to be the same:

Next, we divide both sides by 12:

We arrive at

We’ll be looking at some applications of this very old – and extremely useful – method in future posts.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Literary Terms: What is a Foil?

Tutoring English at the college level, you might be asked to explain the term foil.

There are hundreds of literary terms.  Most of them you never hear in everyday conversation, while many others are seldom used.  In the hallways of universities, however, literary terms are important.

Although most people never discuss them, literary terms can be very interesting.  Today we’ll define and discuss the term foil.

To a jeweler, a foil suggests a thin sheet of metal used as a backing for a gem.  The reflective quality of the foil brightens the gem from behind, enhancing its brilliance.

In literature, a foil is a character who, viewed close to another, emphasizes certain traits of the other character.  The most compelling cases that come to my mind are ones of contrast:  by noticing how good or innocent the foil is, we can’t help but realize how bad or manipulative the more central character is. Of course, the pattern can work the opposite way as well.

In Cinderella, the stepsisters can be seen as foils.  Never working at all, they live in complete self-indulgence.  By contrast, Cinderella works tirelessly to take care of them, never receiving any consideration.  The stepsisters’ selfishness and laziness emphasize how selfless and hardworking Cinderella is.

Another example of a foil might be Banquo from Shakespeare’s MacBeth.  After MacBeth and Banquo encounter the witches, Banquo wants to discuss what they said, but MacBeth himself seems less interested. MacBeth suggests that giving credence to supernatural predictions is unwise.  He seeks to give Banquo the impression that really, what the witches said is not important to him.

In fact, MacBeth murders the King in order to realize one of the witches’ predictions. He goes on to plot the murder of Banquo and Fleance, Banquo’s son, in order to thwart another of the witches’ divinations.

Banquo’s innocent wonder and curiosity about the witches’ predictions, contrasted with MacBeth’s pretended dismissal of them, show us how profoundly the witches affect MacBeth from the very beginning.  In that way, Banquo serves as a foil for MacBeth.

Of course, foil is also a math term.  You can find out about that meaning – among other topics – by searching my math category on the right side of this page:)

Source:  Literary Terms, Coles Notes Study Guide.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Why January can be the hardest month – and how to handle it

Tutoring, you notice which months are tough on students.  There are also your own recollections….

This year, Christmas came a few days after the public school students left for holidays.  As a result, their days off stretched longer at the back end.  Between Christmas and January 7, you get used to a new way of life that doesn’t include going to school.  It can be a hard habit to break.

Remembering my university days, January was always the toughest month for me.  The weather was grey and dismal.  It was so hard to face the cold, grey campus after the festive time of Christmas.  The courses were new, so you weren’t yet engaged with them.  Bottom line:  too many changes at once, against a dismal backdrop.

I found that when February came, I usually felt much better.  The weather was much better by then (of course we’re talking about Victoria, but up here is not too different).  As well, I’d developed some attachment to my courses.  Indeed, February was a much easier month.  Momentum carried me through March – in spite of its avalanche of new material.

Life is about habits.  Good or bad, pleasant or unpleasant, you’ll likely continue a habit.

Here, then, are some of my hints about weathering the month of January:

  1. Don’t expect much from any given day – but go to class every day, anyway.
  2. Remember that 45 minutes of homework is better than nothing, even if it’s not enough.
  3. Try, if possible, to focus on the main idea of what the instructor is discussing.  If you need to throw something away, cut out details.
  4. The days you really don’t want to go to class – but you go anyway – are the most important ones.
  5. Remember:  Everyone else is in the same boat.

Years ago, I recall seeing a student handbook on the ground.  When I picked it up, it opened to a page that showed the principal and vice principal pointing out at the reader.  They were smiling.  The caption read “Remember:  every day counts.”

Those two people understood about habits – and that attending school needs to be one of them.

Good luck this January.  I’ve been there and I know it’s tough.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

English: Your old friends who and whom

Tutoring English, you’re bound to be asked when to use who vs when to use whom.

“Whom do you love?”  asked George Thorogood.  Actually, I think he asked “Who do you love?”  However, he should have used whom.

Whom is for object, whereas who is for subject.  Therefore, the following are correct:

Who loves you?

Whom do you love?

With whomever you discuss who and whom, remember the distinction:  whom for object, who for subject.  However, in a complicated sentence, the object of one verb can be the subject of another.  If it’s a subject, use who – even if it’s also an object of another verb or preposition.

Give the copies to whomever you like.

Give the copies to whoever asks for them.

In the second sentence, the “who” asks for the copies, making that person the subject of “asks”.  Therefore, use “who” rather than “whom”.

How do you tell object and subject apart?  With verbs, the subject does the action, while the object receives the action.  With prepositions, the object follows the preposition.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Blood Sugar Control: The Pancreas, Insulin and Glucagon

Tutoring Biology 12, you talk about the hormones insulin and glucagon, which are used to control blood sugar.

Insulin is much more widely spoken of than glucagon, but in fact they are opposite sides of the blood sugar equation.  Your body uses insulin to lower blood glucose (blood sugar), and glucagon to raise it.  The appropriate blood glucose level is 1 mg/ml, or 0.1%.

When blood sugar rises above normal (just after a meal, for instance), the pancreas releases insulin.  Insulin causes liver, muscle, and fat cells to take in glucose from the blood.  The result:  blood sugar is lowered.  The liver cells convert the glucose to glycogen.  The muscle cells do the same, or else use the glucose to power their efforts if they are working.  The fat cells use the glucose to form fat.  No matter which type of cell takes up the glucose, the glucose gets consumed.

When blood sugar drops below normal (such as between meals), the pancreas releases glucagon.  Glucagon causes the liver to break down glycogen into glucose.  The liver releases that glucose into the bloodstream, raising blood sugar.  Glucagon also causes fat cells to break down fat into glycerol and fatty acids, which then get released in the blood.  The liver takes up the glycerol and fatty acids and converts them into glucose.  The resulting glucose can be released into the blood to increase blood sugar.  Furthermore, glucagon causes the liver to burn fat and protein for its own energy needs, rather than burning glucose.  Hence, more glucose is left in the blood.

Insulin and glucagon are both produced and secreted by sites in the pancreas called the islets of Langerhans.

Source:  Inquiry into Life, 11th Edition, by Sylvia S. Mader.  McGraw-Hill:  2006.

Biology: functions of water in the human body

Tutoring biology 12, the six (or so) functions of water in the human body comprise an early theme.

Everyone knows water is important in the human body, but what are the precise reasons?

1.  Water is liquid at body temperature.  This point is obvious, yet important.

2.  Water is a very effective solvent.  It doesn’t dissolve everything, of course:  if it did, you wouldn’t have a body.  However, water is very good at dissolving salts as well as simple organic molecules that can form H-bonds (sugars, for example).

3.  Water molecules are strongly attracted to each other, so stick together.  Therefore, water beads on a table surface when spilled, and water also can be drawn upwards in a straw.

Water’s tendency to stick together (also called cohesiveness) causes it to move along in a unified manner.  Anything in the water must move with it, since the water molecules pull each other along.  Hence, water is the transport medium  in the human blood system.

4.  Water can release a great deal of heat without cooling down much.  It can also absorb a great deal of heat while only slightly increasing its temperature.  Therefore, water stabilizes the body’s temperature.  As we’ll discuss in later posts, the body works best when its temperature stays at 37°C.

5.  Adding to point 4:  water consumes tremendous energy as it evaporates.  Therefore, sweating cools the body very effectively, guarding against overheating.

6.  Water functions as a lubricant in the body.  In saliva it helps the food move easily down the esophagus.  In synovial fluid it lubricates the knee joint.

Sources:

Inquiry into LIfe, 11th edition, by Sylvia S. Mader.  McGraw-Hill:  2006.

Biology 12, Module 1:  Cell Biology 1.  Open School BC:  2007.