# Topics in Grade 11 Math

• ## Exponent Law for Multiplication

### example question:

Simplify :$$(a)(a^3)$$
• ## Exponent Law for Division

### example question:

Simplify:$$x^7\div x^3$$
• ## Zero Exponent Property

### example question:

Evaluate $$\frac{3^2}{3^2}$$
• ## Negative Exponents

### example question:

Evaluate $$2^{-1}$$
• ## Power Law and Mixed Exponent Practice

### example question:

Simplify $$(a^2)^3$$

### example question:

Evaluate:$$\frac{5^{-4}-5^{-6}}{5^{-3}+5^{-5}}$$  Good test question!!
• ## Rational (Fraction) Exponents (Part 1)

### example question:

Evaluate$$49^\frac{1}{2}$$
• ## Rational (Fraction) Exponents (Part 2)

### example question:

Simplify:$$\sqrt{\sqrt{x^8}}$$
• ## Solving Exponential Equations (Part 1)

### example question:

Solve. $$2^x=8$$
• ## 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?

• ## Simplifying Polynomials Review

### example question:

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

This section is a brief overview.

• ## Factoring Review

### example question:

Factor: $$5y+15$$

This section is a brief overview.

• ## 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:
$$2+6+18+...$$

• ## 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:

### example question:

Simplify $$3\sqrt{11}+5\sqrt{11}+2\sqrt{11}$$
• ## Radicals (Part 2) (Multiplying and Dividing)

### example question:

Simplify: $$\sqrt{50}$$
• ## Radicals (Part 3) (Complex Numbers - Imaginary Numbers "$i$")

### example question:

Simplify: $$\sqrt{-16}$$

### example question:

Simplify $$\frac{2}{\sqrt{3}}$$

### 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.)
• ## Linear - Quadratic Systems

### example question:

Solve: $$y-x=1$$ $$y=x^2-2x-3$$
• ## 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.
$$y=x^2+6x+2$$
• ## 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?

• ## 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:

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. )
• ## Graphing Trigonometric Functions - Degrees

### example question:

Graph the following.
$$y=3\sin 2x$$
• ## Graphing Trigonometric Functions - Radians

### example question:

Graph the following.
$$y=3\sin 2x$$
• ## 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.

$$0$$