# Topics in Grade 11 Math

## Advanced Exponent Work

### example question:

Evaluate:$$\frac{5^{-4}-5^{-6}}{5^{-3}+5^{-5}}$$ $$ $$ Good test question!!## Solving Exponential Equations (Part 2)

### example question:

Solve. $$ 2^{2x}-9(2^x)+8=0$$ $$ $$ Good test question!## Exponential Growth (Doubling Time) $A_{t}=A_{0}(2)^{\frac{t}{d}}$

### example question:

Use the equation $A_{t}=A_{0}(2)^{\frac{t}{d}}$ to solve. $$ $$ If you start with 3 bacteria and the bacterial colony doubles its population every 2 hours, what will the population be after 8 hours?## Exponential Growth (% increase/appreciation) $A_{t}=A_{0}(1+i)^n$

### example question:

Use the equation $A_{t}=A_{0}(1+i)^n$ to solve. $$ $$ The value of a property is expected to increase by 7% per year. If you purchased the property for $100,000, what will it be worth in 12 years?## Exponential Decay (Half-Life) $A_t=A_0(\frac{1}{2})^\frac{t}{h}$

### example question:

Use the equation $A_t=A_0(\frac{1}{2})^\frac{t}{h}$ to solve. $$ $$ A radioactive substance has a half-life of 3 hours. If you start with 100 g, what mass will remain after 12 hours?

(Note: This lesson includes a brief explanation of half-life.)## Exponential Decay (% decrease/depreciation) $A_t=A_0(1-i)^n$

### example question:

Use the equation $A_{t}=A_{0}(1-i)^n$ to solve. $$ $$ A car loses 10% of its value every year. If you purchased the car for $25,000, how much is it worth after 7 years?## Simplifying Polynomials Review

### example question:

Simplify. $$ (3x+5)-(x-4)$$

This section is a brief overview.

For extra review on simpifying polynomials, click here## Factoring Review

### example question:

Factor: $$5y+15$$

This section is a brief overview.

For extra review on common factoring, click here## Multiplying and Dividing Rational Expressions

### example question:

Simplify and state restrictions. $$\frac{x^2+7x+12}{x^2+x-6}\times\frac{x^2+5x-14}{x^2-x-20}$$## Adding and Subtracting Rational Expressions (Part 1)

### example question:

Simplify and state any restrictions. $$ \frac{5}{6x}-\frac{2}{3x}+\frac{3}{4x}$$## Adding and Subtracting Rational Expressions (Part 2)

### example question:

Simplify and state any restrictions. $$ \frac{3}{x-1}+\frac{4}{x-1}$$## Recursion Formulas (Sequences)

### example question:

Given $t_1=3$, Find the first four terms of the sequence determined by:$$ $$ $$t_n=2(t_{n-1})$$## Arithmetic Sequences $t_n=a+(n-1)d$

### example question:

Given the sequence: $$3, 5, 7...$$ Find the 18th term.## Geometric Sequences $t_n=a(r)^{n-1}$

### example question:

Given the sequence: $$ 3,\: 6,\: 12... $$

Find the 7th term.## Arithmetic Series $S_n=\frac{n}{2}[2a+(n-1)d]$ or $S_n=\frac{n}{2}(a+t_n)$

### example question:

Find the sum of the first 40 terms given the arithmetic series:

$$2+5+8+11+...$$## Geometric Series $S_n=\frac{a(r^n-1)}{r-1}$

### example question:

Find $S_8$ for the geometric series:

$$2+6+18+...$$## Financial Math - Simple Interest ($I=Prt$)

### example question:

If you invest $800 at 6% simple interest, how much will you have after 5 years?## Financial Math - Compound Interest - Amount / Present value $A=P(1+\frac{i}{c})^{(tc)}$

### example question:

Calculate the amount if you invest $750 for 8 years at 5.5% compounded annually.## Amount of an Annuity (Annuity "IN") $A=\frac{R[(1+\frac{i}{c})^{tc}-1]}{(\frac{i}{c})}$

### example question:

Calculate the amount of the annuity:

$700 deposited every 6 months for 4 years at 5% compounded semi-annually.## Present Value of an Annuity (Annuity "OUT") $P=\frac{R[1-(1+\frac{i}{c})^{-tc}]}{(\frac{i}{c})}$

### example question:

How much should you deposit into an account now that pays 6% compounded semi-annually so that you can withdraw $700 every 6 months for 4 years? (What is the present value of the annuity?)## Radicals (Part 1) (Adding and Subtracting)

### example question:

Simplify $$3\sqrt{11}+5\sqrt{11}+2\sqrt{11} $$## Radicals (Part 3) (Complex Numbers - Imaginary Numbers "$i$")

### example question:

Simplify: $$\sqrt{-16} $$## Radicals (Part 4) (Rationalizing Denominators) (Radicals and Complex) (Conjugates)

### example question:

Simplify $$\frac{2}{\sqrt{3}} $$## Solving Quadratic Equations (Quadratic Formula, Radicals and Complex / Imaginary Roots)

### example question:

Solve by factoring. $$3x^2+4x-15=0$$## Solving Quadratic Equations (Word Problems)

### example question:

Is it possible to make a rectangle that has a perimeter of 40m and an area of 170$m^2$? (Show work and explain.)## The Discriminant ($b^2-4ac$) (Nature/Number of Roots)

### example question:

If the following equation has 2 real distinct roots, find k. $$kx^2+3x-5=0$$

Important lesson !## Solving for Maximum / Minimum (Complete the Square)

### example question:

Find the maximum or minimum value and the value of $x$ when it occurs.

$$y=x^2+6x+2$$

For extra review on completing the square, click here## Solving for Maximum / Minimum (Word Problems)

### example question:

The equation shows the height (h) of a baseball in metres as a function of time (t) in seconds. $$ h=-5t^2+20t+1$$

a) Find the maximum height of the ball and the time when it occurs.

b) What is the initial height of the ball?

c) When does the ball hit the ground?

For more word problems on completing the square, click here## Function Notation [ $f(x), g(x)$ ] (Functions)

### example question:

Given $f(x)=2x^2-3x+4$, determine the value of:

a) $f(-2)$

b) $f(x-2)$

Good test question !## Inverse of a Function ( $f^{-1}(x)$ )

### example question:

Find the inverse of $f(x)=3x+5$.

( Find $f^{-1}(x)$ )## Graphing $y=|x|$ and its Transformations

### example question:

Graph $y=|x|$, then graph the following: $$y=-2|x-5|-3$$ a) State the domain and range of the transformed graph.

b) State all the transformations.## Graphing $y=\sqrt{x}$ and its Transformations

### example question:

Graph $y=\sqrt{x}$, then graph the following: $$y=-3\sqrt{-2(x+4)}-1$$ a) State the domain and range of the transformed graph.

b) State all the transformations.## Graphing $y=\frac{1}{x}$ and its Transformations

### example question:

Graph $y=\frac{1}{x}$, then graph the following: $$y=\frac{1}{x+4}+5$$ a) State the domain and range of the transformed graph.

b) State all the transformations.## Radians $\Leftrightarrow$ Degrees Conversions (Trigonometry)

### example question:

Convert the following:

a) 30° to radians.

b) $\frac{\pi}{3}$radians to degrees.## Unit Circle, CAST, Quadrants, Exact Values and Special Triangles (Part 1)

### example question:

Point P(3,4) lies on the terminal arm of $\angle{\theta}$ in standard position. ($0°\leq\theta\leq360°$).

a) Determine the exact values for $\sin\theta$, $\cos\theta$, $\tan\theta$, $\csc\theta$, $\sec\theta$, and $\cot\theta$.

b) Find $\theta$ (to the nearest degree).## Unit Circle, CAST, Quadrants, Exact Values and Special Triangles (Part 2)

### example question:

a) Find one negative and one positive co-terminal angle with 30°

b) Find the exact value for sin -420°## Solving Trigonometric Equations - Degrees (Part 1)

### example question:

Solve for $0°\leq x \leq360°$

a) $\sin x=\frac{1}{2}$

b) $\cos x=0$## Solving Trigonometric Equations - Degrees (Part 2)

### example question:

Solve for $0°\leq x \leq360°$

a) $2\sin x+1=0$

b) $2\cos x+1=0$## Solving Trigonometric Equations - Radians (Part 1)

### example question:

Solve for $0 \leq x \leq 2\pi$

a) $\sin x=\frac{1}{2}$

b) $\cos x=0$

( trigonometry / trig )## Solving Trigonometric Equations - Radians (Part 2)

### example question:

Solve for $0 \leq x \leq 2\pi$

a) $2\sin x+1=0$

b) $2\cos x+1=0$## Trigonometric Identities ( Trig Identities )

### example question:

Prove:

$$ \tan x\cos x = \sin x$$ ( Using Trig Identities )## Solving Trigonometric Equations with Identities - Degrees and Radians

### example question:

Solve for $0°\leq x \leq360°$

$$2\sin^2 x+\cos x -1=0$$

( use trig identities. )## Finding the Equations of Trigonometric Functions - Degrees (Part 1)

### example question:

Find the equation of the sin function given:

Amplitude = 5

Period =180°

Phase shift = 30° to the right.

Vertical Shift = 4 units up.## Finding the Equations of Trigonometric Functions - Degrees (Part 2)

### example question:

Find an equation of the trigonometric function given:

Maximum =11 and Minimum = 1

Period = 180°

Is at the maximum value at 0°

## Finding the Equations of Trigonometric Functions - Radians (Part 1)

### example question:

Find the equation of the sin function given:

Amplitude = 5

Period =$\pi$

Phase shift = $\frac{\pi}{6}$ to the right.

Vertical Shift = 4 units up.## Finding the Equations of Trigonometric Functions - Radians (Part 2)

### example question:

Find an equation of the trigonometric function given:

Maximum =10 and Minimum = -1

Period = $\pi$

Is at the maximum value at 0°

## Finding the Equations of Trigonometric Functions - Word Problems (Degrees)

### example question:

Find two equations (using cosine and sine) to model the following:

A mass on the end of a spring is at rest 60 cm above the ground. it is pulled down 40 cm and released at time $t=0$. It takes 2 seconds for the mass to return to the low position.## Finding the Equations of Trigonometric Functions - Word Problems (Radians)

### example question:

Find two equations (using cosine and sine) for the following:

A mass on the end of a spring is at rest 80 cm above the ground. it is pulled down 60 cm and released at time $t=0$. It takes 2 seconds for the mass to return to the low position.

$$0$$