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.
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.
Ready?
10th Grade algebra · Curves and Functions
Quadratics, the quadratic formula, and function notation — the algebra behind every thrown ball and every profit curve.
Full screen — no distractions
For grown-ups: 10th Grade algebra
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.
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.
🏠 Try it at home
f(x) is a machine with a name
Write c(t) for the cost of t cinema tickets. What does c(3) mean in words? Work it out.
Solve by factoring
“I’m thinking of two numbers that multiply to 0.” Whatever they guess, one of them has to be 0. Why?
The quadratic formula
Find the discriminant b² − 4ac for three quadratics. Positive: two answers. Zero: one. Negative: the curve never touches the axis.
The shape of a quadratic
Throw a ball and film it in slow motion. The path is a parabola — the vertex is the highest point.
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.
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.