Grade 11 Math — Functions & Applications

Every parabola turns exactly once, and the turn has coordinates.

Read aloud: k equals b over two, squared. Half FIRST, square second — squaring first is the classic slip, and it misses by a factor of four.

Read aloud: x equals minus b, plus or minus the square root of b-squared minus four a c, all over two a — a formula you can say is a formula you own.

Read aloud: delta equals b-squared minus four a c. Its SIGN answers a question before the formula is even asked: how many real roots are coming.

You can interrogate a quadratic's roots without ever finding them.

Everything this chapter taught now works on the sky: the apex is the vertex, the landing is a root.

Linear things ADD the same amount each step. Exponential things MULTIPLY by the same factor each step — and that one difference is why a savings account, a rumour and a petri dish all behave alike.

The top is the change; the bottom is what you are comparing it WITH. A $20 rise on an $80 drill is a big deal; the same $20 on a $2000 mower is a rounding error.

It is one fact wearing three outfits: solve it for the final amount, for the starting amount, or for the rate, and it answers three different exam questions.

Same machine, different stationery. The named trap of this lesson is adding where you should subtract. A car that loses 20% a year is multiplied by 0.80, not 1.20 — and if your answer is bigger than the sticker price, the sign slipped.

That exponent is doing one simple job — counting how many halvings fit into the elapsed time. So the working move is to count FIRST.

Compare it with the half-life law and you will see the same skeleton: count how many characteristic times fit into the elapsed time, then raise the characteristic factor to that count.

Every question so far handed you the time. Now it is the unknown, and it is stuck up in the exponent where ordinary algebra cannot reach it.

A quantity that repeats — a swing, a tide, a heartbeat — is described by three numbers, and their units keep them apart.

Five cycles each second means each cycle gets a fifth of a second; nothing deeper hides in there.

Run that check on every rearrangement in this lesson: if the units refuse to cancel into the answer's units, the rearrangement is wrong, no appeal.

Length upstairs, gravity downstairs, and the mass nowhere: hang a brass bob or a potato from the same string and they keep identical time.

A Ferris wheel is a periodic function you can ride: a rider's height traces a perfect sinusoid, and behind that curve is plain circular motion.

Five landmarks, and the whole curve is yours.

Every right triangle has three sides, and only one of them has a permanent name.

SOH CAH TOA is not a spell; it is an index. So the pick is mechanical, and it takes five seconds.

This is what trigonometry was invented for: a height nobody can climb, measured from a distance anybody can walk. Stand back a measured distance, sight the top, read the angle — the angle of elevation, taken up from the horizontal.

Take the right angle away and SOH CAH TOA has nothing to hold on to — there is no hypotenuse to be opposite or adjacent TO.

The sine law needs a side paired with the angle facing it. Hand it two sides and the angle WEDGED BETWEEN them and it stalls — there is no matched pair to start from.

What you CAN measure is two sides and the angle between them. So build the height instead.

A sequence is a list with a rule, and this chapter runs on exactly two rules. If each term is made by adding the same amount — 3, 7, 11, 15 — the sequence is arithmetic, and the fixed step is the common difference d.

Here a is the first term and d the common difference. The n - 1 is the whole exam: the first term has taken zero steps, so by term n only n - 1 steps have happened.

Where you will meet it: theatre rows widening toward the back, pipes stacked in narrowing rows, posts, seats, seconds.

Same skeleton as the arithmetic version, same off-by-one, one upgrade: the steps are multiplications now, so they stack as an exponent.

Notice the shape: the LAST term, 384, beats everything before it combined, 381. Multiplying growth is back-loaded — that is its whole personality, and the reason late slices of it dominate every total.

I = Prt — read aloud: I equals P r t. The interest I is what the money earns; P is the principal, the money doing the earning; r the yearly rate; t the years.

Each period multiplies the whole balance by 1 + i, and n periods stack as a power — this is the geometric sequence with r = 1 + i, wearing a bank uniform.

The geometric-series lesson asked you to hold a thought — here is the payoff. Deposit D at the end of every year at rate i: the first deposit compounds for n - 1 years, the next for n - 2, the last not at all.