Warm-up · Algebra Review

🧩 Quadratic Functions Review

Every lesson in this course builds on quadratics. This warm-up reviews the three forms of a quadratic, the vertex, the quadratic formula and completing the square, so the rest of the course has a solid base.

🧾 Three forms of a quadratic

  • Standard form: y = ax² + bx + c — the y-intercept is c.
  • Vertex form: y = a(x - h)² + k — the vertex is (h, k).
  • Factored form: y = a(x - p)(x - q) — the zeros are p and q.

If a > 0 the parabola opens up and the vertex is a minimum; if a < 0 it opens down and the vertex is a maximum. The vertex of standard form is at x = -b / 2a.

🔑 Solving ax² + bx + c = 0

  • Factoring: x² - 5x + 6 = (x - 2)(x - 3), so x = 2 or x = 3.
  • Quadratic formula: x = (-b ± √(b² - 4ac)) / 2a works for every quadratic.
  • Discriminant b² - 4ac: positive → two real roots, zero → one repeated root, negative → no real roots.
  • Completing the square turns standard form into vertex form.
x² + 8x + 3
= (x + 4)² - 16 + 3
= (x + 4)² - 13 → vertex (-4, -13)

🖼️ See it

y = x² − 6x + 5: zeros, vertex and axis of symmetry-101234567-404812x = 3zero x = 1zero x = 5vertex (3, −4)y-intercept 5xy
Factored form (x − 1)(x − 5) gives the zeros, x = −b / 2a gives the axis, and the vertex is the minimum.

📝 Worked examples

Example 1 · Vertex from standard form

y = x² − 6x + 5
x = −b / 2a = 6 / 2 = 3
y = 9 − 18 + 5 = −4
Vertex (3, −4), axis of symmetry x = 3

Example 2 · Quadratic formula

2x² + 3x − 2 = 0 a = 2, b = 3, c = −2
Discriminant = 9 + 16 = 25
x = (−3 ± 5) / 4 → x = 1/2 or x = −2

Example 3 · Completing the square

x² + 8x + 3
= (x + 4)² − 16 + 3
= (x + 4)² − 13 → vertex (−4, −13)

📌 Facts to remember

  • Standard form y = ax² + bx + c shows the y-intercept c.
  • Vertex form y = a(x − h)² + k shows the vertex (h, k).
  • Factored form y = a(x − p)(x − q) shows the zeros p and q.
  • If a > 0 the parabola opens up (minimum); if a < 0 it opens down (maximum).
Discriminant b² − 4ac: positive → two real roots, zero → one repeated root, negative → no real roots.
✏️ Question · 1 of 10
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1. Which is the vertex form of a quadratic?

Vertex form shows (h, k) directly.
✏️ Question · 2 of 10
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2. What is the x-coordinate of the vertex of y = x² - 6x + 5?

Use x = −b / 2a.
✏️ Question · 3 of 10
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3. What is the discriminant of x² + 4x + 5?

Discriminant = b² − 4ac.
✏️ Question · 4 of 10
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4. If the discriminant is negative, the quadratic equation has:

The formula needs √(b² − 4ac).
✏️ Question · 5 of 10
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5. Solve x² - 5x + 6 = 0. (Enter smaller first, separated by comma)

Find two numbers that multiply to 6 and add to −5.
✏️ Question · 6 of 10
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6. Complete the square: x² + 8x + 3 = (x + 4)² + k. What is k?

Half of 8 is 4, and 4² = 16.
✏️ Question · 7 of 10
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7. y = 2(x − 1)² + 7. What is the minimum value of y?

Read k from vertex form.
✏️ Question · 8 of 10
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8. Solve x² − 2x − 8 = 0 with the quadratic formula. What is the larger solution?

b² − 4ac first, then (−b ± √D) / 2a.
✏️ Question · 9 of 10
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9. Which form makes the x-intercepts easiest to read?

Which form has the roots written in it?
✏️ Question · 10 of 10
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10. A ball’s height in metres is h = −5t² + 20t. At what time t, in seconds, is the ball highest?

The highest point is the vertex.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5

The graph of y = −3x² + 2 opens:

Question 2 of 5

The axis of symmetry of y = x² + 4x + 1 is:

Question 3 of 5

A discriminant of 0 means the quadratic has:

Question 4 of 5

The vertex of y = (x − 3)² − 2 is:

Question 5 of 5

The quadratic formula is:

Lesson 1 · Algebra

📈 Polynomial Functions & Zeros

You've mastered quadratics — now it's time to look at polynomials of degree 3 and higher. Their graphs have more turns, but the same core rules of factoring and zeros still apply.

📊 End Behaviour

The "ends" of a polynomial graph are controlled by its degree (even/odd) and leading coefficient (positive/negative).

  • Even degree, positive lead: Both ends point up (like a parabola).
  • Odd degree, positive lead: Left end points down, right end points up.

🎯 Zeros and Multiplicity

The zeros (or roots) are where the graph crosses or touches the x-axis (where y = 0).

  • Multiplicity 1 (odd): The graph crosses straight through the x-axis.
  • Multiplicity 2 (even): The graph bounces off the x-axis and turns around.
y = (x - 2)²(x + 1)
Zero at x = 2 (multiplicity 2 → bounces)
Zero at x = -1 (multiplicity 1 → crosses)

🖼️ See it

y = (x − 2)²(x + 1): bounce and cross-10123-8-404812x = −1 crossesx = 2 bouncesxy
A zero of odd multiplicity crosses the axis; a zero of even multiplicity touches it and turns back.

📝 Worked examples

Example 1 · Find zeros by factoring

Factor out the GCF first, then use the difference of squares.

x³ − 4x = x(x² − 4) = x(x − 2)(x + 2)
Zeros: x = 0, x = 2, x = −2

Example 2 · Sketch from factored form

y = (x + 1)(x − 1)(x − 3)
Zeros: −1, 1, 3 (each crosses)
Degree 3, positive lead → left end down, right end up
y-intercept: (1)(−1)(−3) = 3

Example 3 · Multiplicity

y = x²(x − 4)³
x = 0 has multiplicity 2 → bounces
x = 4 has multiplicity 3 → crosses (and flattens)
Degree = 2 + 3 = 5

📌 Facts to remember

  • The degree equals the sum of the multiplicities.
  • A degree-n polynomial has at most n real zeros and at most n − 1 turning points.
  • The y-intercept is found by plugging in x = 0.
End behaviour: only the leading term matters. −2x⁴ + x behaves like −2x⁴, so both ends go down.
✏️ Question · 1 of 10
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1. A polynomial has an odd degree and a positive leading coefficient. As x → ∞, y goes to:

An odd degree behaves like y = x³ at the ends.
✏️ Question · 2 of 10
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2. What are the zeros of y = (x - 3)(x + 4)? (Enter smaller first, separated by comma)

Set each factor equal to 0.
✏️ Question · 3 of 10
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3. If a polynomial has a factor of (x - 5)³, what does the graph do at x = 5?

Is the multiplicity odd or even?
✏️ Question · 4 of 10
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4. What is the degree of f(x) = 4x⁵ - 2x³ + x - 7?

The degree is the highest power of x.
✏️ Question · 5 of 10
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5. A polynomial has an even degree and a negative leading coefficient. As x → -∞, y goes to:

An even degree with a negative lead behaves like y = −x².
✏️ Question · 6 of 10
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6. What is the y-intercept of f(x) = (x - 2)(x + 3)? (Hint: plug in x = 0)

Substitute x = 0.
✏️ Question · 7 of 10
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7. A polynomial has degree 4. How many real zeros can it have?

Think about the highest power of x.
✏️ Question · 8 of 10
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8. What is the end behaviour of y = −2x⁴ + x?

Look at the leading term: even degree, negative sign.
✏️ Question · 9 of 10
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9. Let y = (x − 1)(x − 4). What is the sum of the zeros?

Set each factor equal to zero.
✏️ Question · 10 of 10
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10. Let y = (x + 1)(x − 1)(x − 3). What is the y-intercept?

Substitute x = 0 into each factor.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5

What is the maximum number of turning points of a degree-5 polynomial?

Question 2 of 5

What are the zeros of y = x²(x + 3)?

Question 3 of 5

At a zero of multiplicity 2, the graph:

Question 4 of 5

Factor x³ − 9x completely.

Question 5 of 5

What is the degree of y = (x − 1)²(x + 2)(x − 5)?

Lesson 2 · Algebra

🧮 Rational Expressions & Equations

Rational expressions are just fractions where the numerator and denominator are polynomials. The golden rule? The denominator can never be zero!

🧹 Simplifying Rational Expressions

Factor the top and bottom, then cancel out any common factors.

(x² - 9) / (x² - 3x)
= (x + 3)(x - 3) / x(x - 3)
= (x + 3) / x   (with x ≠ 3)

✖️➗ Multiplying & Dividing

Multiply straight across (factor first!). To divide, flip the second fraction and multiply.

(x / x+2) ÷ (3 / x+2)
= (x / x+2) × (x+2 / 3)
= x / 3

⚖️ Solving Rational Equations

Multiply every term by the Least Common Denominator (LCD) to clear the fractions, then solve. Always check for extraneous solutions!

2/x = 4/(x + 2)
LCD = x(x + 2)
2(x + 2) = 4x
2x + 4 = 4x → x = 2

🖼️ See it

Simplifying a rational expressionFactorise top andbottom(x² − 9) ÷ (x + 3) = (x− 3)(x + 3) ÷ (x + 3)Cancel commonfactorsleaves x − 3State therestrictionx ≠ −3, since thatmakes the bottom 0
You can only cancel factors, never terms, and the original denominator still cannot be zero.

📝 Worked examples

Example 1 · Adding rational expressions

1/x + 2/(x + 1)
= (x + 1)/(x(x + 1)) + 2x/(x(x + 1))
= (3x + 1) / (x(x + 1))

Example 2 · Solve by cross-multiplying

3/(x − 2) = 6/(x + 1)
3(x + 1) = 6(x − 2)
3x + 3 = 6x − 12
15 = 3x → x = 5

Example 3 · An extraneous solution

x²/(x − 3) = 9/(x − 3)
x² = 9 → x = 3 or x = −3
x = 3 makes the denominator 0 → rejected
Solution: x = −3

📌 Holes vs. asymptotes

After factoring, a factor that cancels leaves a hole. A factor that stays in the denominator gives a vertical asymptote.

(x − 2)(x + 1) / (x − 2)(x + 3)
Hole at x = 2, vertical asymptote at x = −3
Always check your answers against the original denominators. Any value that makes a denominator zero must be thrown out.
✏️ Question · 1 of 10
Question

1. Simplify: (x² - 4) / (x - 2)

Factor the top as a difference of squares.
✏️ Question · 2 of 10
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2. What value of x must be excluded from the domain of 5 / (x - 7)?

The denominator cannot be zero.
✏️ Question · 3 of 10
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3. Multiply and simplify: (2x / 3) × (9 / 4x)

Multiply across, then cancel the x.
✏️ Question · 4 of 10
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4. Solve for x: 3/x = 6/8

Cross-multiply.
✏️ Question · 5 of 10
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5. When solving 1/(x-1) = 2/(x-1) + 1, you might find a solution that makes the denominator zero. This is called an:

It is a solution of the working, not of the original equation.
✏️ Question · 6 of 10
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6. Solve for x: 5/x = 10/4

Cross-multiply.
✏️ Question · 7 of 10
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7. Simplify (x² − 5x + 6) / (x − 2), for x ≠ 2.

Factor the numerator first.
✏️ Question · 8 of 10
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8. Where is the vertical asymptote of y = 1 / (x + 4)?

Set the denominator equal to 0.
✏️ Question · 9 of 10
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9. Solve 6/x = 3/5. What is x?

Cross-multiply: 6 × 5 = 3 × x.
✏️ Question · 10 of 10
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10. Solve 4/(x + 1) = 2. What is x?

Multiply both sides by x + 1.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5

Which values are excluded for 7 / ((x − 1)(x + 2))?

Question 2 of 5

Simplify 3x² / 6x, for x ≠ 0.

Question 3 of 5

What is the LCD of 1/x and 1/(x + 3)?

Question 4 of 5

Solve 2/(x − 1) = 1.

Question 5 of 5

For y = (x − 2)(x + 1) / (x − 2), what happens at x = 2?

Lesson 3 · Algebra

🔁 Introduction to Logarithms

Logarithms are the inverse of exponents. If exponents ask "what do I get when I raise a base to a power?", logarithms ask "what power do I need to raise this base to?"

🔄 Exponential ↔ Logarithmic Form

y = bˣ is the exact same as log_b(y) = x

2³ = 8   ↔   log₂(8) = 3
"Base 2 raised to the 3rd power equals 8."

🔢 Common & Natural Logs

  • Common log (log): Base 10. log(100) = 2
  • Natural log (ln): Base e (≈ 2.718). ln(e²) = 2

📜 Basic Log Properties

  • Product rule: log_b(MN) = log_b(M) + log_b(N)
  • Quotient rule: log_b(M/N) = log_b(M) - log_b(N)
  • Power rule: log_b(M^p) = p × log_b(M)

🖼️ See it

An exponential and its inverse logarithm012345012345y = 2ˣy = log₂ xy = x(1, 2)(2, 1)xy
log₂ x asks “what power of 2 gives x?”. The two graphs are reflections of each other in y = x.

📝 Worked examples

Example 1 · Evaluate by thinking in powers

log₄(64) = ? Ask: 4 to what power is 64?
4³ = 64 → log₄(64) = 3

Example 2 · Combine logs

log(2) + log(50) = log(2 × 50) = log(100) = 2

Example 3 · Expand

log₃(9x²) = log₃(9) + log₃(x²)
= 2 + 2 log₃(x)

Example 4 · Subtract

log₂(32) − log₂(4) = 5 − 2 = 3
(check: log₂(32/4) = log₂(8) = 3)

📌 Special values and limits

  • log_b(1) = 0 because b⁰ = 1.
  • log_b(b) = 1 because b¹ = b.
  • ln(eˣ) = x and e^(ln x) = x.
  • The argument must be positive: log₂(−4) and log₂(0) are undefined.
Not allowed: log(M + N) does not equal log(M) + log(N). The product rule is for log(M × N).
✏️ Question · 1 of 10
Question

1. Rewrite 5² = 25 in logarithmic form.

bʸ = x becomes log_b(x) = y.
✏️ Question · 2 of 10
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2. What is log₃(81)? (Hint: 3 to what power equals 81?)

3 × 3 × 3 × 3 = ?
✏️ Question · 3 of 10
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3. What is log₁₀(1000)?

10 to what power is 1000?
✏️ Question · 4 of 10
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4. Expand log₂(8x) using log properties.

The log of a product is a sum of logs.
✏️ Question · 5 of 10
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5. Rewrite log₃(x⁴) so there are no exponents inside the log.

The power rule moves the exponent to the front.
✏️ Question · 6 of 10
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6. What is ln(e⁵)?

ln and e undo each other.
✏️ Question · 7 of 10
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7. What is log₅(1)?

5 to what power gives 1?
✏️ Question · 8 of 10
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8. Which expression is undefined?

The argument must be positive.
✏️ Question · 9 of 10
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9. Evaluate. What is log₂(32)?

2 × 2 × 2 × 2 × 2 = 32.
✏️ Question · 10 of 10
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10. Evaluate. What is log₁₀(0.01)?

Write 0.01 as 1/100.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5

What is log₉(3)?

Question 2 of 5

Write log₂(5) + log₂(6) as a single logarithm.

Question 3 of 5

What is ln(1)?

Question 4 of 5

What is log₃(81) − log₃(3)?

Question 5 of 5

Expand log(x²y).

Lesson 4 · Algebra

🗝️ Solving Exponential & Log Equations

When the variable is trapped in an exponent, regular algebra won't cut it. You need to use logarithms to "pull it down."

📉 Solving Exponential Equations

Take the log of both sides, then use the power rule to bring the variable down.

Solve: 2ˣ = 10
log(2ˣ) = log(10)
x × log(2) = 1
x = 1 / log(2) ≈ 3.32

🔓 Solving Logarithmic Equations

Use log properties to combine logs, then drop the logs from both sides.

Solve: log₃(x) + log₃(x - 2) = 1
log₃(x(x - 2)) = 1
3¹ = x² - 2x
x² - 2x - 3 = 0
(x - 3)(x + 1) = 0 → x = 3 (x = -1 makes log undefined)

🖼️ See it

Solving 2ˣ = 8 from the graph01234508162432y = 2ˣx = 3, since 2³ = 8xy
Solving 2ˣ = 8 means finding where the curve meets the line y = 8. A logarithm answers the same question: x = log₂ 8 = 3.

📝 Worked examples

Example 1 · Match the bases

4ˣ = 32
(2²)ˣ = 2⁵
2^(2x) = 2⁵ → 2x = 5 → x = 5/2

Example 2 · Isolate the exponent

3^(2x − 1) = 27 = 3³
2x − 1 = 3 → x = 2

Example 3 · Convert a log equation

log₂(x + 3) = 3
x + 3 = 2³ = 8 → x = 5

Example 4 · Doubling time

How long for money to double at 5% per year?

1000(1.05)ᵗ = 2000
(1.05)ᵗ = 2
t = log(2) / log(1.05) ≈ 14.2 years

📌 Strategy

  1. Can both sides be written with the same base? Then equate the exponents.
  2. If not, take a log of both sides and use the power rule.
  3. For log equations, combine into one log, then rewrite in exponential form.
  4. Check every answer: the argument of any log must be positive.
Rejecting answers: log₂(x) + log₂(x − 2) = 3 gives x² − 2x = 8, so x = 4 or x = −2. But x = −2 makes log₂(x) undefined, so only x = 4 works.
✏️ Question · 1 of 10
Question

1. Solve: 5ˣ = 125

Write 125 as a power of 5.
✏️ Question · 2 of 10
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2. To solve 3ˣ = 20, what is the best first step?

20 is not a power of 3, so you need logs.
✏️ Question · 3 of 10
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3. Solve: log₂(x) = 4

Rewrite in exponential form.
✏️ Question · 4 of 10
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4. Solve: log(x) + log(4) = log(20)

log a + log b = log(ab).
✏️ Question · 5 of 10
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5. Solve: 2ˣ⁺¹ = 16

Write 16 as a power of 2.
✏️ Question · 6 of 10
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6. When solving log equations, you must always:

You cannot take the log of zero or a negative number.
✏️ Question · 7 of 10
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7. What is the best first step to solve 7ˣ = 50?

The variable is in the exponent.
✏️ Question · 8 of 10
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8. Solving log₂(x) + log₂(x − 2) = 3 gives x = 4 and x = −2. Which must be rejected?

Which one makes an argument negative?
✏️ Question · 9 of 10
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9. Solve 2^(x + 3) = 64. What is x?

Write 64 as a power of 2.
✏️ Question · 10 of 10
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10. Solve log₃(x − 1) = 2. What is x?

Rewrite as x − 1 = 3².
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5

Solve 10ˣ = 1000.

Question 2 of 5

Solve log₅(x) = 2.

Question 3 of 5

Solve 3^(2x) = 81.

Question 4 of 5

Solve ln(x) = 0.

Question 5 of 5

Solve ln(x) = 2.

Lesson 5 · Geometry

📐 Right Triangle Trigonometry

Trigonometry connects the angles of a right triangle to the ratios of its sides. It's the bridge between algebra and geometry.

📏 SOH CAH TOA

For a right triangle with an angle θ:

  • Sine (sin): sin(θ) = Opposite / Hypotenuse
  • Cosine (cos): cos(θ) = Adjacent / Hypotenuse
  • Tangent (tan): tan(θ) = Opposite / Adjacent

🧭 Solving Right Triangles

If you know one side and one angle (besides the 90°), or two sides, you can find everything else.

Hypotenuse = 10, angle = 30°
Opposite = ?
sin(30°) = Opp / 10
Opp = 10 × sin(30°) = 5

📐 Angles of Elevation & Depression

  • Angle of elevation: Looking UP from the horizontal.
  • Angle of depression: Looking DOWN from the horizontal.

🖼️ See it

A 5-12-13 right triangleadjacent = 12opposite = 5hypotenuse = 13sin θ = 5/13 cos θ = 12/13 tanθ = 5/12
Use SOH CAH TOA with the sides labelled from the angle θ.

📝 Worked examples

Example 1 · Find a missing angle

Opposite = 3, adjacent = 4. Use the inverse tangent.

tan(θ) = 3/4
θ = tan⁻¹(3/4) ≈ 36.9°

Example 2 · The 3-4-5 triangle

Sides 3, 4, 5
sin(θ) = 3/5 cos(θ) = 4/5 tan(θ) = 3/4
(θ is opposite the side of length 3)

Example 3 · Height from an angle of elevation

You stand 30 m from a tower; the angle of elevation is 45°.
tan(45°) = h / 30 → h = 30 × 1 = 30 m

Example 4 · Special triangles

45-45-90: sides 1 : 1 : √2
30-60-90: sides 1 : √3 : 2

📌 Checklist

  1. Draw and label the triangle: hypotenuse, opposite, adjacent for the angle you are using.
  2. Pick the ratio that has your unknown and a known value.
  3. Use inverse trig (sin⁻¹, cos⁻¹, tan⁻¹) to find an angle.
  4. Make sure your calculator is in degree mode.
Pythagorean theorem: a² + b² = c². A 5-12-13 triangle is another common right triangle.
✏️ Question · 1 of 10
Question

1. In a right triangle, Opposite / Hypotenuse is the:

SOH CAH TOA.
✏️ Question · 2 of 10
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2. A right triangle has a hypotenuse of 20 and an angle of 30°. What is the length of the side opposite the 30° angle? (Hint: sin(30°) = 0.5)

opposite = hypotenuse × sin θ.
✏️ Question · 3 of 10
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3. If tan(θ) = 1, what is θ in degrees?

tan θ = 1 when opposite = adjacent.
✏️ Question · 4 of 10
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4. You are standing on a cliff looking DOWN at a boat. The angle between the horizontal and your line of sight is the angle of:

Are you looking up or down?
✏️ Question · 5 of 10
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5. A ladder leans against a wall at a 60° angle. The base is 5 m from the wall. How long is the ladder? (Hint: cos(60°) = 0.5)

hypotenuse = adjacent ÷ cos θ.
✏️ Question · 6 of 10
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6. A right triangle has legs of 7 and 7. What is the measure of one of the acute angles, in degrees?

Equal legs mean an isosceles right triangle.
✏️ Question · 7 of 10
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7. Which ratio equals adjacent ÷ hypotenuse?

SOH CAH TOA.
✏️ Question · 8 of 10
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8. In a 3-4-5 right triangle, what is the sine of the angle opposite the side of length 3?

The hypotenuse is the longest side.
✏️ Question · 9 of 10
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9. You stand 30 m from a tower. The angle of elevation to the top is 45°. How tall is the tower, in metres?

tan θ = opposite ÷ adjacent, and tan 45° = 1.
✏️ Question · 10 of 10
Question

10. A right triangle has legs 6 and 8. What is the length of the hypotenuse?

Use a² + b² = c².
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5

What is sin(30°)?

Question 2 of 5

What is tan(45°)?

Question 3 of 5

A right triangle has hypotenuse 13 and a leg of 5. What is the other leg?

Question 4 of 5

If cos(θ) = 4/5 in a right triangle, what is sin(θ)?

Question 5 of 5

A 10 m pole casts a shadow 10 m long. What is the angle of elevation of the sun?

Lesson 6 · Algebra

🔄 The Unit Circle & Radians

Degrees are great, but radians are the "natural" way to measure angles in higher math. The unit circle ties trigonometry directly to the coordinate plane.

🥧 What is a Radian?

A radian is the angle made when the arc length equals the radius.

  • Full circle = 2π radians = 360°
  • Half circle = π radians = 180°
  • To convert: Degrees × (π / 180) = Radians

🎯 The Unit Circle

A circle with a radius of 1 centreed at the origin (0,0). For any angle θ on the unit circle:

(x, y) = (cos θ, sin θ)

Cosine is the x-coordinate, and sine is the y-coordinate!

📐 Special Triangles

The unit circle is built from two special right triangles:

  • 30-60-90 (gives us π/6 and π/3)
  • 45-45-90 (gives us π/4)

🖼️ See it

Exact values for special anglesDegreesRadianssincostan30°π/61/2√3/21/√345°π/4√2/2√2/2160°π/3√3/21/2√390°π/210undefined
On the unit circle, cos θ is the x-coordinate and sin θ is the y-coordinate. 180° = π radians.

📝 Worked examples

Example 1 · Degrees to radians

Multiply by π/180.

60° × π/180 = π/3
225° × π/180 = 5π/4

Example 2 · Radians to degrees

Multiply by 180/π.

2π/3 × 180/π = 120°

Example 3 · Coordinates on the unit circle

θ = π/6 → (√3/2, 1/2)
θ = π/4 → (√2/2, √2/2)
θ = π/3 → (1/2, √3/2)

Example 4 · Signs by quadrant

Quadrant I: cos +, sin +
Quadrant II: cos −, sin +
Quadrant III: cos −, sin −
Quadrant IV: cos +, sin −

📌 Quick references

  • π/6 = 30°, π/4 = 45°, π/3 = 60°, π/2 = 90°, π = 180°, 3π/2 = 270°, 2π = 360°.
  • The reference angle is the acute angle to the x-axis. Use it to find the value, then fix the sign by quadrant.
Memory trick for the first quadrant: sin goes 1/2, √2/2, √3/2 for 30°, 45°, 60°. Cos goes the opposite way.
✏️ Question · 1 of 10
Question

1. Convert 90° to radians.

180° = π radians.
✏️ Question · 2 of 10
Question

2. On the unit circle, the x-coordinate of a point at angle θ represents the:

A point on the unit circle is (cos θ, sin θ).
✏️ Question · 3 of 10
Question

3. What is cos(0)?

θ = 0 is the point (1, 0).
✏️ Question · 4 of 10
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4. Convert π radians to degrees.

π radians is half a turn.
✏️ Question · 5 of 10
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5. What are the exact (x, y) coordinates on the unit circle at θ = π/2?

π/2 is a quarter turn, straight up.
✏️ Question · 6 of 10
Question

6. What is sin(π)?

θ = π is the point (−1, 0).
✏️ Question · 7 of 10
Question

7. What is 150° in radians?

Multiply by π/180 and simplify.
✏️ Question · 8 of 10
Question

8. What are the coordinates on the unit circle at θ = 3π/2?

3π/2 is 270°.
✏️ Question · 9 of 10
Question

9. Convert 270° to radians. Write π as "pi" (for example 3pi/2). What is 270° in radians?

270° is three quarters of a full turn (2π).
✏️ Question · 10 of 10
Question

10. Convert the angle 2π/3 radians to degrees. How many degrees?

Replace π with 180°.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5

What is sin(π/2)?

Question 2 of 5

What is cos(π)?

Question 3 of 5

What is 45° in radians?

Question 4 of 5

In which quadrant are both sin and cos negative?

Question 5 of 5

What is sin(π/3)?

Lesson 7 · Algebra

🌊 Graphing Trig Functions

Sine and cosine don't just solve triangles — they create beautiful, repeating waves that model everything from sound to tides.

🌊 The Basic Sine & Cosine Waves

y = sin(x) and y = cos(x) are periodic waves that repeat every 2π units.

  • Sine starts at 0, goes up to 1, down to -1, and back.
  • Cosine starts at 1, goes down to -1, and back up.

📏 Amplitude & Period

  • Amplitude: Height from midline to peak. For y = A sin(x), Amplitude = |A|.
  • Period: Length of one full cycle. For y = sin(Bx), Period = 2π / |B|.

📈 Transforming Trig Graphs

y = A sin(B(x - C)) + D

  • A: Amplitude (vertical stretch)
  • B: Affects Period
  • C: Phase shift (horizontal shift)
  • D: Vertical shift (midline)

🖼️ See it

Sine and cosine over one cycle090180270360-1-0.500.51sin xcos xangle (degrees)value
Both waves have amplitude 1 and period 360°. The cosine curve is the sine curve shifted 90° to the left.

📝 Worked examples

Example 1 · Read all the features

y = 2 sin(3x) + 1
Amplitude: 2
Period: 2π/3
Midline: y = 1
Max = 1 + 2 = 3, Min = 1 − 2 = −1

Example 2 · Phase shift

y = cos(x − π/4) → the cosine curve shifted RIGHT by π/4

Example 3 · Write an equation

Amplitude 3, period π, midline y = 0, starts at 0 going up.

B = 2π/period = 2π/π = 2
y = 3 sin(2x)

Example 4 · Five key points of sine

x = 0, π/2, π, 3π/2, 2π
y = 0, 1, 0, −1, 0

📌 Formula sheet

  • Amplitude |A|; period 2π/|B|; midline y = D.
  • Maximum = D + |A|; minimum = D − |A|.
  • A negative A reflects the graph over the midline.
Watch the sign: y = sin(x + π) shifts LEFT by π, not right.
✏️ Question · 1 of 10
Question

1. What is the amplitude of y = 4 sin(x)?

Amplitude is |A| in y = A sin(x).
✏️ Question · 2 of 10
Question

2. What is the period of y = sin(2x)? (Hint: Period = 2π / B)

Period = 2π / B.
✏️ Question · 3 of 10
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3. Where does the graph of y = cos(x) start when x = 0?

cos(0) = 1.
✏️ Question · 4 of 10
Question

4. What is the midline of y = 3 sin(x) + 2?

The midline is the vertical shift D.
✏️ Question · 5 of 10
Question

5. The graph of y = sin(x - π/2) is shifted how far to the right?

sin(x − c) moves the graph right by c.
✏️ Question · 6 of 10
Question

6. Which function family is best for modeling the changing height of the tide over a 24-hour period?

Tides rise and fall in a repeating cycle.
✏️ Question · 7 of 10
Question

7. What is the period of y = cos(x/2)?

Period = 2π ÷ |B| with B = 1/2.
✏️ Question · 8 of 10
Question

8. What is the maximum value of y = 3 sin(x) − 1?

Midline plus amplitude.
✏️ Question · 9 of 10
Question

9. Consider y = −5 cos(x). What is the amplitude?

Amplitude is always positive.
✏️ Question · 10 of 10
Question

10. Consider y = 2 sin(x) + 4. What is the maximum value?

Add the amplitude to the midline.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5

What is the period of y = sin(4x)?

Question 2 of 5

What is the range of y = 2 cos(x)?

Question 3 of 5

What is the midline of y = cos(x) − 3?

Question 4 of 5

How does y = −sin(x) compare to y = sin(x)?

Question 5 of 5

How is y = sin(x + π) shifted from y = sin(x)?

Lesson 8 · Algebra

🎨 Transformations of Functions

You can shift, stretch, shrink, and flip ANY function family using a few simple algebraic rules applied to the function's equation.

🔀 The Master Transformation Formula

y = a * f(b(x - h)) + k

Outside the function (affects y):

  • +k moves UP, -k moves DOWN.
  • a > 1 stretches vertically, 0 < a < 1 shrinks vertically.
  • Negative a flips it over the x-axis.

Inside the function (affects x):

  • -h moves RIGHT, +h moves LEFT.
  • b > 1 shrinks horizontally, 0 < b < 1 stretches horizontally.

📝 Example

Start with f(x) = x².
Move it right 2, up 3, and flip it over the x-axis:
y = -(x - 2)² + 3

🖼️ See it

Transforming y = f(x)Outside the functionf(x) + k: up k unitsf(x) − k: down k units−f(x): reflect in the x-axisa·f(x): stretch verticallyInside the functionf(x + h): left h unitsf(x − h): right h unitsf(−x): reflect in the y-axisf(ax): squash horizontally
Changes inside the brackets act on x and do the opposite of what you might expect.

📝 Worked examples

Example 1 · Absolute value

y = |x + 3| − 2
Left 3, down 2 → vertex at (−3, −2)

Example 2 · Reflection and stretch

y = −2 f(x)
Flip over the x-axis AND stretch vertically by a factor of 2.

Example 3 · Track a point

The point (3, 5) is on y = f(x). Where is it on y = f(x − 2) + 1?

x: 3 + 2 = 5 (right 2)
y: 5 + 1 = 6 (up 1)
New point: (5, 6)

Example 4 · Horizontal changes work backwards

y = f(2x): x-coordinates are HALVED. (4, 6) → (2, 6)

📌 Rules of thumb

  • Outside the function: changes y, and does what you expect (+ goes up).
  • Inside the function: changes x, and does the opposite (x − 3 goes right).
  • f(−x) reflects over the y-axis. −f(x) reflects over the x-axis.
Stretch and shrink: for y = ½f(x), the graph is squashed vertically by a factor of ½. For f(2x), it is squeezed horizontally.
✏️ Question · 1 of 10
Question

1. If you add 5 to a function, like y = f(x) + 5, the graph shifts:

Adding outside f moves the graph vertically.
✏️ Question · 2 of 10
Question

2. How is the graph of y = f(x - 4) transformed from y = f(x)?

Inside changes do the opposite of what they look like.
✏️ Question · 3 of 10
Question

3. Multiplying a function by -1, like y = -f(x), reflects the graph across the:

Every y-value changes sign.
✏️ Question · 4 of 10
Question

4. If y = 3f(x), the graph is:

Multiplying outside f changes the heights.
✏️ Question · 5 of 10
Question

5. Describe the transformation: y = f(x + 2) - 5

Inside: horizontal, opposite sign. Outside: vertical.
✏️ Question · 6 of 10
Question

6. If f(x) = x², what is the equation for f(x) shifted right 3 and up 1?

Right 3 means x − 3 inside; up 1 means + 1 outside.
✏️ Question · 7 of 10
Question

7. Which transformation reflects the graph of f over the y-axis?

The sign must go on the x inside the function.
✏️ Question · 8 of 10
Question

8. How does y = ½ f(x) compare with y = f(x)?

The ½ is outside the function.
✏️ Question · 9 of 10
Question

9. The point (3, 5) lies on y = f(x). Consider y = f(x − 2) + 1. What is the y-coordinate of the image of the point?

Only the "+ 1" outside affects y.
✏️ Question · 10 of 10
Question

10. The point (3, 5) lies on y = f(x). Consider y = f(x − 2) + 1. What is the x-coordinate of the image of the point?

Inside changes go the opposite way.
📝 Lesson quiz · 5 questions
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Re-read the lesson, then give it another go.
Question 1 of 5

How does y = f(x) − 3 compare with y = f(x)?

Question 2 of 5

How does y = f(x + 6) compare with y = f(x)?

Question 3 of 5

What is the vertex of y = (x + 1)² − 4?

Question 4 of 5

What does y = f(2x) do to the graph?

Question 5 of 5

A point (4, −1) is reflected over the x-axis. What are its new coordinates?

Lesson 9 · Data

🎲 Conditional Probability

Sometimes the probability of an event changes depending on what already happened. That's conditional probability.

🔗 Conditional Probability Formula

P(A | B) means "the probability of A, given that B has already happened."

P(A | B) = P(A and B) / P(B)

📋 Using Two-Way Tables

Two-way tables are perfect for conditional probability. Just restrict your sample space to the "given" condition.

CategoryLikes MathLikes ScienceTotal
Grade 9201030
Grade 10152540
Total353570

P(Likes Math | Grade 10) = 15 / 40 = 0.375

⚖️ Independent vs. Dependent

  • Independent events: P(A | B) = P(A). The condition changes nothing.
  • Dependent events: P(A | B) ≠ P(A). The condition changes the outcome.

🖼️ See it

A two-way table for conditional probabilityLikes teaNo teaTotalBoys12820Girls181230Total302050
P(likes tea | girl) = 18 ÷ 30 = 0.6. The condition “girl” shrinks the sample space to the girls’ row.

📝 Worked examples

Example 1 · From a two-way table

Using the table above (Grade 9: 20 math, 10 science; Grade 10: 15 math, 25 science):

P(Math | Grade 9) = 20 / 30 = 2/3
P(Grade 10 | Science) = 25 / 35 = 5/7

Example 2 · Using the formula

P(A and B) = 0.12, P(B) = 0.3
P(A | B) = 0.12 / 0.3 = 0.4

Example 3 · Test for independence

P(A) = 0.5, P(B) = 0.4, P(A and B) = 0.2
P(A) × P(B) = 0.2 = P(A and B) → independent

Example 4 · Without replacement

Two aces in a row from a deck:

P(1st ace) × P(2nd ace | 1st ace) = 4/52 × 3/51 = 1/221

📌 Key ideas

  • Multiplication rule: P(A and B) = P(B) × P(A | B).
  • "Given B" means your new sample space is only B.
  • P(A | B) and P(B | A) are usually different. For example P(rain | clouds) is not P(clouds | rain).
Independent events satisfy P(A and B) = P(A) × P(B).
✏️ Question · 1 of 10
Question

1. P(A | B) means:

Read the bar as “given”.
✏️ Question · 2 of 10
Question

2. Using the table above, what is P(Likes Science | Grade 9)? (Express as decimal to 3 places or 1/3)

Use only the Grade 9 row.
✏️ Question · 3 of 10
Question

3. If events A and B are independent, what is P(A | B)?

Independent means knowing B changes nothing.
✏️ Question · 4 of 10
Question

4. Drawing two cards from a deck without replacement is an example of:

Does the first card change what is left?
✏️ Question · 5 of 10
Question

5. A bag has 3 red and 2 blue marbles. You draw one blue marble and DON'T replace it. What is the probability the next marble is blue?

After one blue is taken, count what is left.
✏️ Question · 6 of 10
Question

6. If P(Rain | Clouds) = 0.8, this means:

The condition comes after the bar.
✏️ Question · 7 of 10
Question

7. Which statement is generally true?

The "given" event is the new sample space.
✏️ Question · 8 of 10
Question

8. From the table above, what is P(Likes Science | Grade 10)?

Grade 10 has 40 students in total.
✏️ Question · 9 of 10
Question

9. P(A and B) = 0.12 and P(B) = 0.3. What is P(A | B)? (Enter a decimal)

Divide the joint probability by P(B).
✏️ Question · 10 of 10
Question

10. A bag has 5 red and 3 blue marbles. You draw a red marble and keep it. What is the probability the next marble is red? (Enter a fraction)

After the first draw, there are 7 marbles left.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5

P(B) = 0.5 and P(A | B) = 0.6. What is P(A and B)?

Question 2 of 5

A die is rolled and shows more than 3. What is the probability it is even?

Question 3 of 5

Events A and B are independent if:

Question 4 of 5

Using the table above, what is P(Grade 9 | Likes Math)?

Question 5 of 5

You flip a fair coin twice. What is P(heads on the second flip | heads on the first)?

Lesson 10 · Geometry

🍕 Circles, Arcs, & Sectors

Circles have their own special measurements. When you slice a circle like a pizza, you create arcs and sectors that rely on radians and proportions.

📏 Arc Length

The distance along the curved edge of a circle.

s = r * θ (θ MUST be in radians!)

Or using degrees: Arc length = (θ / 360) * 2πr

🍕 Sector Area

A sector is a "slice" of a circle. Its area is a fraction of the whole circle's area.

Sector Area = (1/2) * r² * θ (θ in radians)

Or using degrees: Sector Area = (θ / 360) * πr²

📐 Inscribed Angles

An inscribed angle has its vertex ON the circle.

The Inscribed Angle Theorem: An inscribed angle is exactly HALF the measure of its intercepted arc.

🖼️ See it

Arcs and sectorsQuantityDegreesRadiansArc lengthθ/360 × 2πrrθSector areaθ/360 × πr²½r²θ
A sector is a fraction θ/360 of a whole circle, so its arc and area are the same fraction of the circumference and area.

📝 Worked examples

Example 1 · Arc length with degrees

r = 9, θ = 40°
Arc = (40/360) × 2π × 9 = (1/9) × 18π = 2π

Example 2 · Sector area with radians

r = 6, θ = π/3
Area = ½ r² θ = ½ × 36 × π/3 = 6π

Example 3 · Inscribed angle on a diameter

An angle inscribed in a semicircle intercepts an arc of 180°, so it is 90°.

Example 4 · Find the central angle

r = 4, arc length s = 2π
s = rθ → θ = 2π/4 = π/2

📌 Formula sheet

  • Circumference C = 2πr, area A = πr².
  • Arc length s = rθ (radians) or (θ/360) × 2πr (degrees).
  • Sector area ½r²θ (radians) or (θ/360) × πr² (degrees).
  • Inscribed angle = ½ × intercepted arc = ½ × the central angle on the same arc.
Units: the shortcuts s = rθ and A = ½r²θ only work when θ is in radians.
✏️ Question · 1 of 10
Question

1. Which formula calculates arc length when θ is in radians?

Arc length is radius times angle in radians.
✏️ Question · 2 of 10
Question

2. A circle has a radius of 6. What is the arc length of a central angle of π/3 radians?

s = rθ.
✏️ Question · 3 of 10
Question

3. What is the area of a sector with radius 4 and central angle π/2 radians?

Sector area = ½ r² θ.
✏️ Question · 4 of 10
Question

4. If an intercepted arc measures 120°, what is the measure of the inscribed angle in degrees?

An inscribed angle is half its intercepted arc.
✏️ Question · 5 of 10
Question

5. A sector has a central angle of 90°. What fraction of the whole circle's area is the sector?

What fraction of 360° is the angle?
✏️ Question · 6 of 10
Question

6. A central angle measures 80°. What is the measure of an inscribed angle that intercepts the SAME arc?

The inscribed angle is half the central angle on the same arc.
✏️ Question · 7 of 10
Question

7. An inscribed angle intercepts a semicircle. Its measure is:

Half the intercepted arc.
✏️ Question · 8 of 10
Question

8. A circle has radius 5. What is the arc length for a central angle of 2 radians?

Use s = rθ.
✏️ Question · 9 of 10
Question

9. A circle has radius 8 and a central angle of π/4 radians. Write π as "pi", for example 3pi. What is the arc length?

Multiply 8 by π/4.
✏️ Question · 10 of 10
Question

10. A sector has radius 10 and a central angle of 90°. Write π as "pi", for example 3pi. What is its area?

90° is a quarter of a full circle.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5

What is the circumference of a circle with radius 7?

Question 2 of 5

What is the area of a sector with r = 3 and θ = 2π/3?

Question 3 of 5

A central angle measures 100°. What is the inscribed angle on the same arc?

Question 4 of 5

A circle has radius 4 and an arc of length 2π. What is the central angle?

Question 5 of 5

What is the area of a circle with radius 5?

Assessment

🏆 Final Challenge Quiz

Test your overall mastery across all Grade 10 Math Adventure units!

0 / 10
💪 Keep practicing — go back through the lessons and retake the quiz!
Q1

Q1. What is the end behaviour of f(x) = -x³ + 2?

Q2

Q2. Simplify: (x² - 9) / (x + 3)

Q3

Q3. What is log₂(16)?

Q4

Q4. Solve: 3ˣ = 81

Q5

Q5. In a right triangle, what ratio is Opposite / Adjacent?

Q6

Q6. What is the exact value of sin(π/6)?

Q7

Q7. What is the period of y = cos(3x)?

Q8

Q8. The graph of y = f(x - 2) + 4 is shifted:

Q9

Q9. If P(A and B) = 0.2 and P(B) = 0.5, what is P(A | B)?

Q10

Q10. What is the arc length of a circle with radius 10 and a central angle of π/2 radians?