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10th Grade algebra · Ages 15–16

Curves and Functions

Quadratics, the quadratic formula, and function notation — the algebra behind every thrown ball and every profit curve.

f(x)the unknown

The one idea

A function is a rule with a name. A quadratic is the rule that makes a curve — and its zeros are where the curve meets the ground.

Grade 10 asks for fluency rather than new ideas. Solving a quadratic by factoring is the Grade 9 rewrite plus one fact: if two things multiply to zero, one of them is zero. The quadratic formula is the fallback when factoring is hard — not something to fear.

📚 The 4 lessons✏️ Practice

This page is the right one if they can already…

  • Factors x² + bx + c
  • Graphs lines from y = mx + b
  • Uses exponent rules confidently

Not yet? Start at 9th Grade: Algebra 1 — algebra only works one rung at a time.

The four 10th Grade algebra lessons

In order — each one leans on the last. Every example is broken into single moves and ends by checking itself.

1

f(x) is a machine with a name

f(x) = 3x − 2 is the Grade 2 number machine, named f. f(4) means “put 4 into machine f”.

Worked example

f(x) = 3x − 2. Find f(4).

  1. 1

    Replace every x with 4.

    f(4) = 3(4) − 2

  2. 2

    Multiply.

    12 − 2

  3. 3

    Subtract.

    f(4) = 10

Answer: f(4) = 10

Check: Machine f: triple, then take 2. Triple 4 is 12, take 2 is 10 ✓

⚠️ The usual mistake

Reads f(4) as f × 4.

Brackets have always meant multiply until now.

Fix: Read it aloud as “f of 4” — the output of machine f when 4 goes in.

💬 Say this

“What does machine f do? Put 4 in and run it.”

🏠 Try it at home

Write c(t) for the cost of t cinema tickets. What does c(3) mean in words? Work it out.

2

Solve by factoring

Factor, then use one fact: if two things multiply to zero, at least one of them must be zero.

−5−555xy
  • y = x² − 5x + 6
  • zeros: 2, 3
  • vertex (2.5, −0.25)

Worked example

x² − 5x + 6 = 0

  1. 1

    Find two numbers that multiply to 6 and add to −5.

    −2 and −3

  2. 2

    Factor.

    (x − 2)(x − 3) = 0

  3. 3

    One bracket must be zero.

    x − 2 = 0 or x − 3 = 0 → x = 2 or x = 3

Answer: x = 2 or x = 3

Check: 2² − 5(2) + 6 = 0 ✓ and 3² − 5(3) + 6 = 0 ✓ — the curve crosses the x-axis at 2 and 3.

⚠️ The usual mistake

Divides both sides by x and loses an answer, or gives only one answer.

Every equation before this year had one answer.

Fix: A curve that dips below the axis crosses it twice. Expect two answers and check both.

💬 Say this

“If two numbers multiply to make zero, what do you know about them?”

🏠 Try it at home

“I’m thinking of two numbers that multiply to 0.” Whatever they guess, one of them has to be 0. Why?

3

The quadratic formula

When factoring is hard, the formula always works: x = (−b ± √(b² − 4ac)) ÷ 2a.

−5−555xy
  • y = 2x² + 3x − 2
  • zeros: −2, 0.5
  • vertex (−0.75, −3.12)

Worked example

2x² + 3x − 2 = 0

  1. 1

    Read off a, b and c.

    a = 2, b = 3, c = −2

  2. 2

    Work out the part under the root.

    b² − 4ac = 9 + 16 = 25, √25 = 5

  3. 3

    Use + and − in turn.

    x = (−3 + 5) ÷ 4 = ½ or x = (−3 − 5) ÷ 4 = −2

Answer: x = ½ or x = −2

Check: 2(½)² + 3(½) − 2 = ½ + 1½ − 2 = 0 ✓

⚠️ The usual mistake

Gets b² − 4ac = 9 − 16 = −7.

c is negative, and −4 × 2 × −2 has two minus signs that are easy to lose.

Fix: Put every value in brackets: (3)² − 4(2)(−2). Two negatives multiply to a positive.

💬 Say this

“Write a, b and c down first — with their signs.”

🏠 Try it at home

Find the discriminant b² − 4ac for three quadratics. Positive: two answers. Zero: one. Negative: the curve never touches the axis.

4

The shape of a quadratic

y = (x − h)² + k is a U shape with its lowest point at (h, k). A minus in front flips it upside down.

−5−555xy
  • y = (x − 2)² − 1
  • zeros: 1, 3
  • vertex (2, −1)

Worked example

Where is the lowest point of y = (x − 2)² − 1?

  1. 1

    A square is never negative, so (x − 2)² is smallest when it is 0.

    x − 2 = 0 → x = 2

  2. 2

    Put x = 2 in.

    y = 0 − 1 = −1

  3. 3

    That is the vertex.

    (2, −1)

Answer: The vertex is (2, −1).

Check: Expand: (x − 2)² − 1 = x² − 4x + 3, which factors to (x − 1)(x − 3) — zeros at 1 and 3, either side of 2 ✓

⚠️ The usual mistake

Says the vertex is at x = −2.

They read the −2 inside the bracket as it is written.

Fix: Ask what makes the bracket zero. x − 2 is zero when x is +2.

💬 Say this

“What value of x makes the squared part as small as possible?”

🏠 Try it at home

Throw a ball and film it in slow motion. The path is a parabola — the vertex is the highest point.

✏️ Practice that never runs out

Every question is new. A wrong answer gets a hint; a second one shows the worked solution, so no one stays stuck.

Solve the quadratic

Factor, then each bracket can be zero. Tap both answers.

Solve. Tap both answers.

x² − 2x − 8 = 0

Evaluate the function

Replace every x with the number, then work it out.

Find f(1).

f(x) = x² + x + 5

The words, in plain English

Function notation
f(x) names a rule; f(4) is its output for 4.
f(x) = 3x − 2, f(4) = 10
Zero (root)
An x value that makes the function 0.
x² − 5x + 6 has zeros 2 and 3.
Parabola
The U-shaped graph of a quadratic.
y = x²
Vertex
The turning point of a parabola.
(2, −1)
Discriminant
b² − 4ac — tells you how many zeros.
25 → two zeros

🌍 Algebra in an ordinary day

  • The arc of a basketball shot or a fountain
  • Profit that rises, peaks, then falls as the price goes up
  • Spreadsheet formulas are functions: =3*A1−2

🏁 By the end of 10th Grade

  • Reads and evaluates function notation
  • Solves quadratics by factoring and gets both answers
  • Uses the quadratic formula and the discriminant
  • Finds the vertex and sketches the parabola

Where this fits

One idea, twelve rungs. The hiding spot just gets cleverer.

  1. Pre-K❓The Hiding Game
  2. K🎁Make It Fair
  3. 1□The Same As
  4. 2?Number Machines
  5. 3nLetters Are Boxes
  6. 4nNumber Recipes
  7. 5xUndo It
  8. 6xBalance and Tip
  9. 7xTwo Steps Back
  10. 8y = mx + bLines and Slope
  11. 9xAlgebra 1
  12. 10f(x)Curves and Functions

← One rung down

9th Grade: Algebra 1

All 12 gradesMore maths, grade by grade →

Frequently Asked Questions

What algebra should a 10th Grade child learn?

A function is a rule with a name. A quadratic is the rule that makes a curve — and its zeros are where the curve meets the ground. By the end of the year they should be able to: reads and evaluates function notation; solves quadratics by factoring and gets both answers; uses the quadratic formula and the discriminant; finds the vertex and sketches the parabola.

Isn’t 10th Grade too early for algebra?

No — curves and functions is standard 10th Grade maths in most curricula. This page breaks it into single moves so it never feels like a jump.

What is the most common 10th Grade algebra mistake?

Divides both sides by x and loses an answer, or gives only one answer. Every equation before this year had one answer. The fix: A curve that dips below the axis crosses it twice. Expect two answers and check both.

What if my child is stuck?

Go down one rung. Every lesson here builds on the 9th Grade page, and the “ready for this page if” list at the top tells you exactly what they need first. In practice, a wrong answer gives a hint, and a second wrong answer shows the full worked solution.

Is it free?

Yes. All 4 lessons, the pictures, the worked examples, the balance scale and the practice questions are free with no signup. The practice never runs out — every question is generated fresh and checked.

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