Quadratic Functions
Quadratics make parabolas with the form y = ax² + bx + c.
About this topic
A quadratic function has the form f(x) = ax² + bx + c with a ≠ 0, and its graph is a U-shaped curve called a parabola. If a is positive it opens upward, if negative it opens downward, and the larger the size of a, the narrower the curve.
The parabola has an axis of symmetry and a turning point called the vertex, which is the minimum or maximum value. For f(x) = x² − 4x + 3 the axis is x = 2, the vertex is (2, −1), and the zeros are 1 and 3. The constant c is the y-intercept, since f(0) = c.
Quadratics model anything with a single turning point: the path of a thrown basketball, the shape of satellite dishes and car headlights, bridge cables and arches, and revenue that rises and then falls as price changes. The reflective property of the parabola sends parallel rays to a single focus.
A common mistake is to think the graph is V-shaped or has pointed edges; it is smooth and symmetric. Another is to forget that a cannot be zero, since then the function becomes linear. Always check whether the question wants the vertex, the intercepts or the maximum value.
Key ideas
- a decides opening direction and width
- c is the y-intercept
- The vertex is the turning point
Quick check
Use cases: where Quadratic Functions helps
- Homework: review the key idea behind Quadratic Functions before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Quadratic Functions without notes.
- Visual learning: play with the interactive lab to see how Quadratic Functions changes when you adjust its inputs.
- Teaching a friend: explain Quadratic Functions in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Quadratic Functions?
Quadratics make parabolas with the form y = ax² + bx + c.
What will I learn in Quadratic Functions?
a decides opening direction and width. c is the y-intercept. The vertex is the turning point.
Is Quadratic Functions part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Quadratic Functions lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Quadratic Functions?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.