🧩 What is a polynomial?
Each piece separated by a + or − sign is called a term. The degree of a polynomial is the highest exponent on its variable.
4x³ + 2x − 7
Terms: 4x³, 2x, −7
Degree: 3 (from the x³ term)
Lesson 1 · Algebra
🧮 Polynomials
A polynomial is an expression built from variables, coefficients, and exponents, combined with addition, subtraction, and multiplication. Adding, subtracting, and multiplying them are core algebra skills.
➕ Adding & subtracting polynomials
Combine like terms — terms with the exact same variable part.
(3x + 5) + (2x − 3)
= (3x + 2x) + (5 − 3)
= 5x + 2
✖️ Multiplying polynomials
Distribute every term in the first polynomial across every term in the second (often remembered as FOIL for two binomials: First, Outer, Inner, Last).
(x + 2)(x + 3)
= x·x + x·3 + 2·x + 2·3
= x² + 3x + 2x + 6
= x² + 5x + 6
🖼️ See it
📝 Worked examples
Example 1 · Subtracting polynomials
Subtract (5x² + 3x − 4) − (2x² − x + 6). The minus sign in front of the second bracket flips every sign inside it.
(5x² + 3x − 4) − (2x² − x + 6)
= 5x² + 3x − 4 − 2x² + x − 6
= (5x² − 2x²) + (3x + x) + (−4 − 6)
= 3x² + 4x − 10
Example 2 · Multiplying with a negative
(2x − 3)(x + 4)
= 2x·x + 2x·4 − 3·x − 3·4
= 2x² + 8x − 3x − 12
= 2x² + 5x − 12
Example 3 · Squaring a binomial
(x + 3)² means (x + 3)(x + 3) — it is not x² + 9.
(x + 3)(x + 3)
= x² + 3x + 3x + 9
= x² + 6x + 9
⚠️ Names and common mistakes
Polynomials are named by their number of terms: monomial (1 term), binomial (2 terms), trinomial (3 terms). Standard form lists terms from the highest power down, and the leading coefficient is the number on the first term.
An expression is not a polynomial if a variable has a negative or fractional exponent, or sits in a denominator, such as 5x⁻¹ or 3/x.
Watch out: when subtracting, forgetting to flip all the signs in the second bracket is the #1 mistake. Write the minus-distribution as its own step.
✏️ Quick check · 1 of 10
Find the degree
What is the degree of 4x³ + 2x − 7?
The degree is the highest power of x.
✏️ Quick check · 2 of 10
Add polynomials
Simplify: (3x + 5) + (2x − 3)
Add the x-terms together and the numbers together.
✍️ Type your answer · 3 of 10
Evaluate the polynomial.
What is 2x² + 3x − 1 when x = 2?
Enter an exact answer, without rounding.Put 2 in for x: work out x² first, then multiply and add.
✏️ Quick check · 4 of 10
Multiply binomials
Multiply: (x + 2)(x + 3)
FOIL: First, Outer, Inner, Last, then combine the middle terms.
✍️ Type your answer · 5 of 10
Evaluate the polynomial.
What is 5x − 4 when x = −3?
Enter an exact answer, without rounding.Put −3 in for x: 5 × (−3) first, then subtract 4.
✍️ Type your answer · 6 of 10
Evaluate the polynomial.
What is 3x² + 5x + 2 when x = 1?
Enter an exact answer, without rounding.Put 1 in for x: every power of 1 is 1.
✏️ Quick check · 7 of 10
Name that polynomial
What is 7x² − 4 called?
Count the terms separated by + or − signs.
✏️ Quick check · 8 of 10
Spot the non-polynomial
Which expression is NOT a polynomial?
Look for a variable with a negative exponent or in a denominator.
✍️ Type your answer · 9 of 10
Subtract (5x² + 3x − 4) − (2x² − x + 6).
What is the coefficient of x in the answer?
Enter an exact answer, without rounding.Distribute the minus sign: −(−x) becomes +x.
✍️ Type your answer · 10 of 10
Multiply (x + 3)(x − 5).
What is the coefficient of x in the product?
Enter an exact answer, without rounding.The two middle terms are −5x and +3x. Combine them.
📝 Lesson quiz · 5 questions
Question 1 of 5
What is the degree of 6x⁴ − 2x² + 9?
Question 2 of 5
Simplify (4x² + x) + (3x² − 5x).
Question 3 of 5
Multiply (x − 4)(x + 4).
Question 4 of 5
Expand (x + 5)².
Question 5 of 5
What is the leading coefficient of 3x − 8x³ + 2?
Lesson 2 · Algebra
✂️ Factoring Polynomials
Factoring rewrites a polynomial as a product of simpler pieces — it's multiplying in reverse, and it's the key that unlocks solving quadratic equations.
🔑 Factoring out the GCF
Find the greatest common factor (GCF) of every term, then pull it out front.
6x + 9
GCF of 6 and 9 is 3
= 3(2x + 3)
🧮 Factoring trinomials
For x² + bx + c, find two numbers that multiply to c and add to b.
x² + 7x + 12
Need two numbers: multiply to 12, add to 7 → 3 and 4
= (x + 3)(x + 4)
⚡ Difference of squares
a² − b² = (a + b)(a − b)
x² − 25
= x² − 5²
= (x + 5)(x − 5)
🖼️ See it
📝 Worked examples
Example 1 · Factor a trinomial
Factor x² + 7x + 12. Find two numbers that multiply to 12 and add to 7: 3 and 4.
x² + 7x + 12 = (x + 3)(x + 4)
Check: x² + 4x + 3x + 12 = x² + 7x + 12 ✓
Example 2 · Negative constant
Factor x² + 2x − 15. Multiply to −15, add to +2: 5 and −3.
x² + 2x − 15 = (x + 5)(x − 3)
Example 3 · GCF first, then factor
3x² + 12x + 9
= 3(x² + 4x + 3) ← pull out the GCF 3
= 3(x + 1)(x + 3)
🔎 A checklist for any factoring problem
- Pull out the greatest common factor (GCF) first.
- Two terms? Look for a difference of squares: a² − b² = (a + b)(a − b).
- Three terms? Find two numbers that multiply to the constant and add to the middle coefficient.
- Always multiply back to check.
Sign trick: if the constant is positive, both numbers share the sign of the middle term. If the constant is negative, the numbers have opposite signs and the bigger one takes the middle term's sign.
✏️ Quick check · 1 of 10
Factor out the GCF
Factor: 6x + 9
What is the biggest number that divides both 6 and 9?
✏️ Quick check · 2 of 10
Factor a trinomial
Factor: x² + 7x + 12
Find two numbers that multiply to 12 and add to 7.
✍️ Type your answer · 3 of 10
Find the greatest common factor.
What is the GCF of 12 and 18?
Enter an exact answer, without rounding.List the factors of each number and find the biggest one they share.
✏️ Quick check · 4 of 10
Factor a difference of squares
Factor: x² − 25
a² − b² = (a + b)(a − b). What squared is 25?
✍️ Type your answer · 5 of 10
Factor x² + 9x + 20 by finding two numbers that multiply to 20 and add to 9.
Enter the smaller of the two numbers.
Enter an exact answer, without rounding.Try the factor pairs of 20: 1 × 20, 2 × 10, 4 × 5.
✍️ Type your answer · 6 of 10
The GCF of 8x² and 12x is 4x.
What is the coefficient (just the number part) of that GCF?
Enter an exact answer, without rounding.The number part of 4x.
✏️ Quick check · 7 of 10
Find the factors
Factor x² + 7x + 12.
Which two numbers multiply to 12 and add to 7?
✏️ Quick check · 8 of 10
Difference of squares
Factor x² − 49.
Both terms are perfect squares, separated by a minus sign.
✍️ Type your answer · 9 of 10
Factor 5x² + 10x. The GCF is a·x.
What is the number a?
Enter an exact answer, without rounding.What is the biggest number dividing both 5 and 10?
✍️ Type your answer · 10 of 10
Factor x² + 2x − 15 as (x + p)(x − 3).
What is p?
Enter an exact answer, without rounding.p × (−3) must equal −15.
📝 Lesson quiz · 5 questions
Question 1 of 5
Factor x² + 9x + 20.
Question 2 of 5
Factor x² − 5x + 6.
Question 3 of 5
Factor 4x² − 25.
Question 4 of 5
Factor 6x² + 9x completely.
Question 5 of 5
Which is the complete factorisation of 2x² − 18?
Lesson 3 · Algebra
🎯 Solving Quadratic Equations
A quadratic equation can have up to two solutions. Factoring is often the fastest method, but the quadratic formula always works, even when factoring doesn't.
🎯 Solving by factoring
Once a quadratic is factored, use the zero product property: if two things multiply to zero, at least one of them must be zero.
(x − 3)(x + 2) = 0
x − 3 = 0 or x + 2 = 0
x = 3 or x = −2
📏 The quadratic formula
For any equation in the form ax² + bx + c = 0, this formula always finds the solutions.
x = (−b ± √(b² − 4ac)) / 2a
🔍 The discriminant
The part under the square root, b² − 4ac, is called the discriminant. It tells you how many real solutions exist.
Positive discriminant
Two real solutions.
Negative discriminant
No real solutions.
🖼️ See it
📝 Worked examples
Example 1 · Solve by factoring
x² − 5x + 6 = 0
(x − 2)(x − 3) = 0
x − 2 = 0 or x − 3 = 0
x = 2 or x = 3
This works because of the zero product property: if two things multiply to 0, at least one of them is 0.
Example 2 · Rearrange first
Solve x² = 4x + 5. The equation must equal 0 before you factor.
x² − 4x − 5 = 0
(x − 5)(x + 1) = 0
x = 5 or x = −1
Example 3 · The quadratic formula
Solve 2x² + 3x − 2 = 0 with a = 2, b = 3, c = −2.
x = (−b ± √(b² − 4ac)) / 2a
= (−3 ± √(9 + 16)) / 4
= (−3 ± 5) / 4
x = 2/4 = 1/2 or x = −8/4 = −2
🔍 The discriminant tells you how many solutions
Compute D = b² − 4ac before you do anything else.
- D > 0 → two real solutions
- D = 0 → exactly one real solution
- D < 0 → no real solutions (the parabola never touches the x-axis)
Remember: never divide both sides by x. You lose the solution x = 0. Move everything to one side and factor instead.
✍️ Type your answer · 1 of 10
Solve by factoring: (x − 3)(x + 2) = 0
Enter the positive solution.
Enter an exact answer, without rounding.If a product is 0, one of the brackets must be 0.
✏️ Quick check · 2 of 10
Recall the quadratic formula
The quadratic formula is:
The formula starts with −b and has ± in front of the square root.
✍️ Type your answer · 3 of 10
For x² − 5x + 6 = 0 (a = 1, b = −5, c = 6):
What is the discriminant, b² − 4ac?
Enter an exact answer, without rounding.(−5)² − 4 × 1 × 6.
✏️ Quick check · 4 of 10
Interpret the discriminant
If the discriminant is negative, the quadratic equation has:
You cannot take the square root of a negative number.
✍️ Type your answer · 5 of 10
Solve: x² − 9 = 0
Enter the positive solution.
Enter an exact answer, without rounding.Add 9 to both sides, then take the square root.
✍️ Type your answer · 6 of 10
Solve by factoring: x² + 2x − 8 = 0
Enter the positive solution.
Enter an exact answer, without rounding.Find two numbers that multiply to −8 and add to 2.
✏️ Quick check · 7 of 10
Zero product property
Solve (x + 4)(x − 1) = 0.
Set each bracket equal to 0 separately.
✏️ Quick check · 8 of 10
How many solutions?
How many real solutions does x² + 2x + 5 = 0 have?
Compute b² − 4ac with a = 1, b = 2, c = 5.
✍️ Type your answer · 9 of 10
For x² − 6x + 5 = 0, the discriminant is b² − 4ac.
What is the discriminant?
Enter an exact answer, without rounding.a = 1, b = −6, c = 5. Square b first; (−6)² is positive.
✍️ Type your answer · 10 of 10
Solve x² − 8x + 15 = 0. There are two solutions.
What is the larger solution?
Enter an exact answer, without rounding.Find two numbers that multiply to 15 and add to −8.
📝 Lesson quiz · 5 questions
Question 1 of 5
Solve x² − 9 = 0.
Question 2 of 5
Solve x² + x − 6 = 0.
Question 3 of 5
What is the discriminant of 2x² − 4x + 2?
Question 4 of 5
In the quadratic formula, what goes under the square root?
Question 5 of 5
Solve x² = 3x.
Lesson 4 · Algebra
🎰 Quadratic Functions & Parabolas
A quadratic function graphs as a U-shaped curve called a parabola. Its shape, direction, and turning point are all controlled by the numbers in the equation.
🌞 Opening up or down
y = ax² + bx + c
a > 0
The parabola opens up (like a smile).
a < 0
The parabola opens down (like a frown).
📍 Finding the vertex
The vertex is the parabola's highest or lowest point. Its x-coordinate always follows this formula:
x = −b / 2a
y = x² − 6x + 5
x = −(−6) / 2(1) = 6/2 = 3
📏 Axis of symmetry & y-intercept
The parabola is a mirror image across a vertical line through the vertex, called the axis of symmetry. The y-intercept is simply c — plug in x = 0.
y = 3x² + 2x − 7
y-intercept = c = −7
🖼️ See it
📝 Worked examples
Example 1 · Vertex and axis of symmetry
For y = x² − 6x + 5, the axis of symmetry is x = −b / 2a.
x = −(−6) / (2·1) = 3
y = 3² − 6(3) + 5 = 9 − 18 + 5 = −4
Vertex: (3, −4)
Example 2 · Intercepts
y-intercept: set x = 0 → y = 5
x-intercepts: x² − 6x + 5 = 0
(x − 1)(x − 5) = 0 → x = 1 and x = 5
Example 3 · Which way does it open?
y = −2x² + 8x: here a = −2, which is negative, so the parabola opens down and the vertex is the maximum point.
x = −8 / (2·−2) = 2
y = −2(4) + 8(2) = 8
Maximum at (2, 8)
📌 Quick facts about parabolas
- a > 0 opens up and has a minimum. a < 0 opens down and has a maximum.
- A bigger |a| makes the parabola narrower; a smaller |a| makes it wider.
- The y-intercept is always the constant c.
- The vertex lies on the axis of symmetry, exactly halfway between the x-intercepts.
Shortcut: if you know the two x-intercepts, the axis of symmetry is their average. For x = 1 and x = 5 that is x = 3.
✏️ Quick check · 1 of 10
Direction of opening
In y = 2x² − 4x + 1, does the parabola open up or down?
Look at the sign of the x² term.
✍️ Type your answer · 2 of 10
For y = x² − 6x + 5:
What is the x-coordinate of the vertex?
Enter an exact answer, without rounding.Vertex x = −b ÷ 2a.
✏️ Quick check · 3 of 10
Read the y-intercept
What is the y-intercept of y = 3x² + 2x − 7?
Put x = 0: only the constant is left.
✍️ Type your answer · 4 of 10
For y = x² − 4x + 3:
What is the x-coordinate of the vertex?
Enter an exact answer, without rounding.Vertex x = −b ÷ 2a.
✏️ Quick check · 5 of 10
Direction with a negative leading coefficient
A parabola with a < 0 opens:
A negative a flips the parabola over.
✍️ Type your answer · 6 of 10
For y = x² − 6x + 5, the vertex is at x = 3.
What is y at x = 3?
Enter an exact answer, without rounding.Put 3 in for x: 9 − 18 + 5.
✏️ Quick check · 7 of 10
Opens which way?
Which parabola opens downward?
Look at the coefficient of x².
✏️ Quick check · 8 of 10
Reading the y-intercept
What is the y-intercept of y = 2x² − 5x + 7?
Substitute x = 0.
✍️ Type your answer · 9 of 10
For y = x² − 4x + 1, use x = −b / 2a.
What is the x-coordinate of the vertex?
Enter an exact answer, without rounding.Here a = 1 and b = −4.
✍️ Type your answer · 10 of 10
Find the vertex of y = x² − 6x + 5. The x-coordinate is 3.
What is the y-coordinate of the vertex?
Enter an exact answer, without rounding.Plug x = 3 into the equation.
📝 Lesson quiz · 5 questions
Question 1 of 5
What is the axis of symmetry of y = x² + 8x + 3?
Question 2 of 5
y = −x² + 4 has its vertex at:
Question 3 of 5
The x-intercepts of y = x² − 9 are:
Question 4 of 5
A parabola has x-intercepts at x = 2 and x = 8. Where is its axis of symmetry?
Question 5 of 5
Which parabola is the narrowest?
Lesson 5 · Algebra
🚀 Exponential Functions
Exponential functions model quantities that grow or shrink by the same percentage over and over — like populations, investments, or radioactive decay.
📈 Growth & 📉 decay
Exponential growth
y = a(1 + r)t
Exponential decay
y = a(1 − r)t
a is the starting amount, r is the rate (as a decimal), and t is time.
🐇 Exponential vs. linear growth
Linear growth adds the same amount every step. Exponential growth multiplies by the same factor every step — which eventually grows much faster.
Linear: 2, 4, 6, 8, 10 (add 2 each time)
Exponential: 2, 4, 8, 16, 32 (double each time)
🔢 Evaluating exponential functions
f(x) = 2x
f(4) = 2⁴ = 16
🖼️ See it
📝 Worked examples
Example 1 · Compound growth
A town of 2,000 people grows 5% each year. After t years: y = 2000(1.05)ᵗ.
After 1 year: 2000 × 1.05 = 2100
After 2 years: 2000 × 1.05² = 2205
Example 2 · Decay
A car worth 20,000 loses 10% of its value each year: y = 20000(0.9)ᵗ.
After 2 years: 20000 × 0.9² = 20000 × 0.81 = 16200
Example 3 · Reading the formula
In y = a·bᵗ, a is the starting amount and b is the multiplier each step. If b > 1 it is growth; if 0 < b < 1 it is decay.
⚖️ Linear vs. exponential
Linear functions add the same amount each step. Exponential functions multiply by the same factor each step. Check a table: constant differences mean linear, constant ratios mean exponential.
x: 0 1 2 3
y: 3 6 12 24 ratios 2, 2, 2 → exponential (y = 3·2ˣ)
Percent to multiplier: grow 8% → multiply by 1.08. Shrink 8% → multiply by 0.92. Never use 0.08 for growth.
✍️ Type your answer · 1 of 10
A population starts at 100 and grows 10% per year.
What is the population after 1 year?
Enter an exact answer, without rounding.Find 10% of 100 and add it on.
✏️ Quick check · 2 of 10
Recall the decay formula
Exponential decay uses which formula?
Decay means losing a percentage, so subtract r inside the brackets.
✍️ Type your answer · 3 of 10
A car worth $20,000 depreciates 15% per year.
What is its value after 1 year?
Enter an exact answer, without rounding.dollars
Find 15% of $20,000 and take it off.
✏️ Quick check · 4 of 10
Compare growth types
Which grows faster in the long run: exponential or linear growth?
Multiplying each step overtakes adding each step.
✍️ Type your answer · 5 of 10
Evaluate the function.
What is f(x) = 2x when x = 4?
Enter an exact answer, without rounding.Multiply 2 by itself 4 times.
✍️ Type your answer · 6 of 10
A population of 500 doubles every year.
What is the population after 2 years?
Enter an exact answer, without rounding.Double 500, then double again.
✏️ Quick check · 7 of 10
Growth or decay?
Which function shows exponential decay?
Look at the base being raised to the power x.
✏️ Quick check · 8 of 10
Pick the multiplier
A price rises 6% per year. What multiplier do you use each year?
Growth multiplier = 1 + rate as a decimal.
✍️ Type your answer · 9 of 10
Evaluate f(x) = 3·2ˣ.
What is f(4)?
Enter an exact answer, without rounding.Calculate 2⁴ first, then multiply by 3.
✍️ Type your answer · 10 of 10
A population of 500 doubles every year: P = 500·2ᵗ.
What is the population after 3 years?
Enter an exact answer, without rounding.Compute 2³ = 8.
📝 Lesson quiz · 5 questions
Question 1 of 5
Which table shows exponential growth?
Question 2 of 5
What is the starting value of y = 7(1.2)ˣ?
Question 3 of 5
A value of 1000 loses 20% per year. After 2 years it is:
Question 4 of 5
Evaluate 5·3² .
Question 5 of 5
In y = a·bˣ, what does b = 1 give?
Lesson 6 · Algebra
√ Radicals & Rational Exponents
Radicals (square roots and beyond) and rational exponents are two ways of writing the same idea. Simplifying them cleanly makes later algebra much easier.
🧩 Simplifying radicals
Pull out any perfect-square factor hiding inside the radical.
√50 = √(25 × 2) = √25 × √2 = 5√2
➕ Operations with radicals
You can only add or subtract like radicals — the same number under the root.
2√3 + 5√3 = 7√3
🔢 Rational exponents
a1/n = n√a
161/2 = √16 = 4
271/3 = ³√27 = 3
🖼️ See it
📝 Worked examples
Example 1 · Simplify √72
Pull out the largest perfect square factor: 72 = 36 × 2.
√72 = √36 · √2 = 6√2
Example 2 · Add radicals
You can only add radicals with the same radicand (the number inside).
3√5 + 4√5 = 7√5
√12 + √3 = 2√3 + √3 = 3√3 ← simplify first!
Example 3 · Rational exponents
x^(1/2) means √x. In general x^(m/n) = (ⁿ√x)ᵐ.
27^(2/3) = (³√27)² = 3² = 9
16^(1/2) = 4
8^(−1/3) = 1 / 8^(1/3) = 1/2
📌 Radical rules to memorise
- √a · √b = √(ab)
- √(a/b) = √a / √b
- √(a²) = |a|
- x^(−n) = 1 / xⁿ
Common trap: √(a + b) is NOT √a + √b. For example √(9 + 16) = √25 = 5, but √9 + √16 = 7.
✍️ Type your answer · 1 of 10
Simplify √50 to the form a√2.
What is a?
Enter an exact answer, without rounding.50 = 25 × 2, and √25 = 5.
✏️ Quick check · 2 of 10
Simplify a radical
√18 simplifies to:
18 = 9 × 2, and √9 = 3.
✍️ Type your answer · 3 of 10
Add the like radicals.
What is 2√3 + 5√3? Enter just the coefficient (the number in front of √3).
Enter an exact answer, without rounding.Like radicals add like terms: 2 + 5.
✏️ Quick check · 4 of 10
Evaluate a rational exponent
161/2 equals:
A power of 1/2 means the square root.
✍️ Type your answer · 5 of 10
Evaluate the expression.
What is 271/3?
Enter an exact answer, without rounding.A power of 1/3 means the cube root: what cubed is 27?
✍️ Type your answer · 6 of 10
Simplify √20 to the form a√5.
What is a?
Enter an exact answer, without rounding.20 = 4 × 5, and √4 = 2.
✏️ Quick check · 7 of 10
Simplify the radical
Simplify √50.
Find the biggest perfect square that divides 50.
✏️ Quick check · 8 of 10
Add radicals
Simplify 2√3 + 5√3.
Treat √3 like a variable, such as 2x + 5x.
✍️ Type your answer · 9 of 10
Evaluate 25^(1/2).
What is the value?
Enter an exact answer, without rounding.x^(1/2) is the same as √x.
✍️ Type your answer · 10 of 10
Evaluate 8^(2/3).
What is the value?
Enter an exact answer, without rounding.Take the cube root first, then square the result.
📝 Lesson quiz · 5 questions
Question 1 of 5
Simplify √48.
Question 2 of 5
Which equals 9^(1/2)?
Question 3 of 5
Simplify √2 · √8.
Question 4 of 5
What is 2^(−3)?
Question 5 of 5
Evaluate 64^(1/3).
Lesson 7 · Algebra
🔀 Functions: Domain, Range & Notation
Function notation is a compact way to describe input-output relationships. Domain and range describe every value that's allowed to go in, and every value that can come out.
🔢 Function notation
f(x) means "the output of function f when the input is x." Substitute the input for x and simplify.
f(x) = 3x + 2
f(5) = 3(5) + 2 = 15 + 2 = 17
📥 Domain & 📤 range
Domain
All possible input (x) values.
Range
All possible output (y) values.
📈 The vertical line test
A graph represents a function only if every vertical line crosses it at most once — meaning every input has exactly one output.
🖼️ See it
📝 Worked examples
Example 1 · Function notation
If f(x) = 3x − 2, then f(5) means "replace x with 5".
f(5) = 3(5) − 2 = 13
f(−1) = 3(−1) − 2 = −5
Example 2 · Domain restrictions
The domain is every input that works. Two things break a function: dividing by zero and taking the square root of a negative number.
g(x) = 1/(x − 4) → x ≠ 4
h(x) = √(x + 3) → x + 3 ≥ 0, so x ≥ −3
Example 3 · Is it a function?
A relation is a function if every input has exactly one output.
{(1,2), (2,4), (3,6)} → function
{(1,2), (1,5), (2,3)} → NOT a function (input 1 has two outputs)
🧭 Domain vs. range
Domain = all possible inputs (x-values). Range = all resulting outputs (y-values). On a graph, read the domain left to right and the range bottom to top.
Vertical line test: if any vertical line crosses a graph more than once, it is not a function.
✍️ Type your answer · 1 of 10
Evaluate the function.
If f(x) = 3x + 2, what is f(5)?
Enter an exact answer, without rounding.Replace x with 5: 3 × 5 + 2.
✏️ Quick check · 2 of 10
Define domain
The domain of a function is:
The domain is what you are allowed to put in.
✍️ Type your answer · 3 of 10
Evaluate the function.
If g(x) = x² − 1, what is g(4)?
Enter an exact answer, without rounding.Replace x with 4: 4² − 1.
✏️ Quick check · 4 of 10
Test for a function
Which test determines if a graph represents a function?
Each input can have only one output.
✍️ Type your answer · 5 of 10
Evaluate the function.
If h(x) = 2x² + 1, what is h(−2)?
Enter an exact answer, without rounding.Square −2 first (it becomes positive), then double and add 1.
✏️ Quick check · 6 of 10
Define range
The range of a function is:
The range is what comes out.
✏️ Quick check · 7 of 10
Function or not?
Which set of pairs is a function?
Check whether any x-value appears twice.
✏️ Quick check · 8 of 10
Find the restriction
What value of x is NOT allowed in f(x) = 5/(x + 2)?
Set the denominator equal to 0.
✍️ Type your answer · 9 of 10
Let f(x) = 2x² − 3.
What is f(3)?
Enter an exact answer, without rounding.Square 3 first, then multiply by 2.
✍️ Type your answer · 10 of 10
Let g(x) = 4x + 1.
For what x does g(x) = 25?
Enter an exact answer, without rounding.Set up the equation 4x + 1 = 25 and solve it.
📝 Lesson quiz · 5 questions
Question 1 of 5
If f(x) = x + 7, what is f(−3)?
Question 2 of 5
What is the domain of h(x) = √(x − 2)?
Question 3 of 5
What is the range of y = x² ?
Question 4 of 5
A circle on a graph fails the vertical line test because:
Question 5 of 5
If f(x) = 3x and g(x) = x + 1, what is f(g(2))?
Lesson 8 · Algebra
🔢 Sequences
A sequence is an ordered list of numbers following a rule. The two most common types — arithmetic and geometric — differ in how they step from one term to the next.
➕ Arithmetic sequences
Each term is found by adding the same number, the common difference (d), to the term before it.
an = a1 + (n − 1)d
3, 7, 11, 15, ...
Common difference: d = 4
✖️ Geometric sequences
Each term is found by multiplying by the same number, the common ratio (r).
an = a1 × rn−1
2, 6, 18, 54, ...
Common ratio: r = 3
🔍 Finding a term
Arithmetic: 3, 7, 11, 15, ...
5th term = 3 + 4×4 = 3 + 16 = 19
Geometric: 5, 10, 20, ...
4th term = 5 × 2³ = 5 × 8 = 40
🖼️ See it
📝 Worked examples
Example 1 · Arithmetic sequence
3, 7, 11, 15, … adds 4 each time. The common difference is d = 4. The rule is aₙ = a₁ + (n − 1)d.
a₁₀ = 3 + (10 − 1)(4) = 3 + 36 = 39
Example 2 · Geometric sequence
2, 6, 18, 54, … multiplies by 3 each time. The common ratio is r = 3. The rule is aₙ = a₁ · rⁿ⁻¹.
a₅ = 2 · 3⁴ = 2 · 81 = 162
Example 3 · Which type?
5, 10, 15, 20 → differences all 5 → arithmetic
5, 10, 20, 40 → ratios all 2 → geometric
🔑 How to decide quickly
- Subtract neighbouring terms. Same answer every time → arithmetic.
- Divide neighbouring terms. Same answer every time → geometric.
- Neither → it is some other pattern.
Off-by-one trap: the nth term uses (n − 1), not n. To get from the 1st term to the 10th you take 9 steps.
✍️ Type your answer · 1 of 10
In the arithmetic sequence 3, 7, 11, 15, ...
What is the common difference?
Enter an exact answer, without rounding.Subtract one term from the next.
✍️ Type your answer · 2 of 10
In the arithmetic sequence 3, 7, 11, 15, ...
What is the 5th term?
Enter an exact answer, without rounding.Add the common difference to the 4th term.
✏️ Quick check · 3 of 10
Name the sequence type
A sequence with a common ratio is called:
Arithmetic adds the same amount; the other type multiplies.
✍️ Type your answer · 4 of 10
In the geometric sequence 2, 6, 18, 54, ...
What is the common ratio?
Enter an exact answer, without rounding.Divide one term by the term before it.
✍️ Type your answer · 5 of 10
In the geometric sequence 5, 10, 20, ...
What is the 4th term?
Enter an exact answer, without rounding.Multiply the 3rd term by the ratio.
✏️ Quick check · 6 of 10
Recall the arithmetic formula
The formula for the nth term of an arithmetic sequence is:
Start at a₁ and add d once for every step after the first.
✏️ Quick check · 7 of 10
Which type?
What kind of sequence is 4, 12, 36, 108, …?
Divide each term by the one before it.
✏️ Quick check · 8 of 10
Next term
What is the next term in 20, 17, 14, 11, …?
Find the difference between neighbours.
✍️ Type your answer · 9 of 10
Arithmetic sequence: a₁ = 5, d = 3.
What is a₈?
Enter an exact answer, without rounding.Use aₙ = a₁ + (n − 1)d with n = 8.
✍️ Type your answer · 10 of 10
Geometric sequence: a₁ = 3, r = 2.
What is a₆?
Enter an exact answer, without rounding.Use aₙ = a₁ · rⁿ⁻¹ with n = 6, so the exponent is 5.
📝 Lesson quiz · 5 questions
Question 1 of 5
What is the common difference of 9, 5, 1, −3, …?
Question 2 of 5
What is the 4th term of aₙ = 2n + 1?
Question 3 of 5
What is the common ratio of 80, 40, 20, 10, …?
Question 4 of 5
Which sequence is arithmetic?
Question 5 of 5
Find a₇ in the arithmetic sequence 2, 5, 8, 11, …
Lesson 9 · Algebra
📍 Absolute Value Equations & Inequalities
Absolute value measures distance from zero, so it's always non-negative. That one idea — distance can't be negative — is the key to solving these equations.
📏 What absolute value means
|x| = distance from 0
|−15| = 15
|15| = 15
Absolute value strips away the sign, so the result is always zero or positive.
⚖️ Solving absolute value equations
Since the expression inside could have been positive or negative before the absolute value was applied, split into two cases.
|x − 3| = 5
x − 3 = 5 or x − 3 = −5
x = 8 or x = −2
↔️ Absolute value inequalities
|x| < a
x is between −a and a.
|x| > a
x is beyond −a or a (less than −a, or greater than a).
🖼️ See it
📝 Worked examples
Example 1 · Basic equation
|x| = 5 means x is 5 units from zero, so x = 5 or x = −5.
Example 2 · Isolate first
|2x − 1| + 3 = 10
|2x − 1| = 7 ← isolate the absolute value
2x − 1 = 7 or 2x − 1 = −7
x = 4 or x = −3
Example 3 · No solution
|x + 2| = −4 has no solution because an absolute value can never be negative.
📌 The 3-step method
- Isolate the absolute value on one side.
- If the other side is negative → no solution. Otherwise split into two equations: inside = positive value, inside = negative value.
- Solve both, then check each answer in the original.
Distance idea: |x − 3| = 4 means "x is 4 away from 3", so x = 7 or x = −1.
✍️ Type your answer · 1 of 10
Find the absolute value.
What is |−15|?
Enter an exact answer, without rounding.Absolute value is the distance from zero.
✍️ Type your answer · 2 of 10
Solve: |x| = 7
Enter the positive solution.
Enter an exact answer, without rounding.Which two numbers are 7 away from 0?
✏️ Quick check · 3 of 10
Split into two cases
Solving |x − 3| = 5 gives two equations: x − 3 = 5 and x − 3 = −5. What are the two solutions?
Solve each equation separately: add 3 to both sides.
✍️ Type your answer · 4 of 10
Solve: |x + 4| = 10
Enter the positive solution.
Enter an exact answer, without rounding.x + 4 = 10 or x + 4 = −10.
✏️ Quick check · 5 of 10
Interpret an inequality
|x| < 5 means:
Less than 5 away from zero, on either side.
✍️ Type your answer · 6 of 10
Solve: |2x| = 16
Enter the positive solution.
Enter an exact answer, without rounding.2x = 16 or 2x = −16.
✏️ Quick check · 7 of 10
Solve it
Solve |x − 3| = 4.
Make two equations: x − 3 = 4 and x − 3 = −4.
✏️ Quick check · 8 of 10
Any solutions?
How many solutions does |x| + 5 = 2 have?
Isolate |x| first.
✍️ Type your answer · 9 of 10
Solve |x + 2| = 9. One solution is x = 7.
What is the other solution?
Enter an exact answer, without rounding.Set x + 2 equal to −9.
✍️ Type your answer · 10 of 10
Solve |3x| = 12. There are two solutions.
What is the positive solution?
Enter an exact answer, without rounding.Divide 12 by 3.
📝 Lesson quiz · 5 questions
Question 1 of 5
What is |−8| ?
Question 2 of 5
Solve |x| = 0.
Question 3 of 5
Solve |2x| = 10.
Question 4 of 5
Solve |x − 1| + 2 = 6.
Question 5 of 5
|x − 6| = 2 means x is 2 units from:
Final Challenge
🏆 19-Question Challenge Quiz
One question from every topic you just learned. Pick an answer for each question, then hit Check My Answers.
Q1 · Polynomials
What is the degree of 5x⁴ − 3x + 1?
Q2 · Factoring
Factor: x² − 36
Q3 · Quadratic Equations
Solve x² − 4 = 0. What is the positive solution?
Q4 · Quadratic Functions
For y = x² + 4x + 3, does the parabola open up or down?
Q5 · Exponential Functions
A population of 200 grows 20% per year. What is it after 1 year?
Q6 · Radicals
√32 simplifies to:
Q7 · Functions
If f(x) = 4x − 1, what is f(3)?
Q8 · Sequences
What is the common ratio of 3, 9, 27, 81?
Q9 · Absolute Value
What is |−22|?
Q10 · Quadratics
The quadratic formula is used to solve equations in the form:
Q11 · Polynomials
What is the leading coefficient of 3x − 8x³ + 2?
Q12 · Factoring Polynomials
Which is the complete factorisation of 2x² − 18?
Q13 · Solving Quadratic Equations
Solve x² = 3x.
Q14 · Quadratic Functions & Parabolas
Which parabola is the narrowest?
Q15 · Exponential Functions
In y = a·bˣ, what does b = 1 give?
Q16 · Radicals & Rational Exponents
Evaluate 64^(1/3).
Q17 · Functions: Domain, Range & Notation
If f(x) = 3x and g(x) = x + 1, what is f(g(2))?
Q18 · Sequences
Find a₇ in the arithmetic sequence 2, 5, 8, 11, …
Q19 · Absolute Value Equations & Inequalities
|x − 6| = 2 means x is 2 units from: