Quadratic function
Quadratic function - theory
A quadratic function describes a relationship that includes x². Its graph is a parabola, and the formula itself lets you read a lot about the shape and position of the graph.
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What is a quadratic function?
A quadratic function is a function that can be written as f(x) = ax² + bx + c,where a, b, and c are real numbers and a ≠ 0.
The condition a ≠ 0 is essential: if a were equal to 0, the term with x² would disappear and we would get a linear function.
shape
direction and width of the arms
position
affects the vertex
start
Y-axis intercept
general form
y = ax² + bx + c
Example: f(x) = x² - 2x - 3
x = -1
f(-1) = (-1)² - 2 · (-1) - 3 = 0
This is one of the roots of the parabola.
x = 1
f(1) = 1² - 2 · 1 - 3 = -4
This is the vertex, the lowest point of this graph.
x = 3
f(3) = 3² - 2 · 3 - 3 = 0
The second root lies symmetrically on the other side.
Vertex and axis of symmetry
The vertex is the turning point of the parabola. A vertical axis of symmetry passes through this point and divides the graph into two mirror-image parts.
Next step
Once you know the general idea, go to the three forms of a quadratic function formula or try a few examples of your own in the calculator.