Introduction:
A quadratic function is a type of polynomial function that has a degree of 2. It is generally represented as:
$$ f(x) = ax^2 + bx + c $$
Where, a, b, c are constanat numbers and a ≠ 0.
Here are some key properties of quadratic functions, along with examples:
Parabola Shape
If \( a > 0 \), the parabola opens upward; if \( a < 0 \), it opens downward.
Vertex
The vertex is the highest or lowest point of the parabola, depending on the direction it opens.
The vertex is at \( x = \frac{-b}{2a} \). Substitute this into the function to find the y-coordinate, that is,
\( y = \frac{4ac - b^2}{4a}\)
Axis of Symmetry
A quadratic function is symmetric about a vertical line called the axis of symmetry.
The axis of symmetry is \( x = \frac{-b}{2a} \).
Roots/Zeros
The roots (or zeros) are the x-values where the parabola intersects the x-axis (f(x) = 0), that is,
$$ ax^2 + bx + c = 0 $$
Roots are found using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
The discriminant (Δ = b² - 4ac) determines the nature of the roots:
- Δ > 0: Two distinct real roots.
- Δ = 0: One real root (the vertex lies on the x-axis).
- Δ < 0: No real roots (the parabola does not cross the x-axis)
Y-Intercept
The y-intercept is the point where the parabola crosses the y-axis (x = 0).
The y-intercept is \( c \) (where \( x = 0 \)).
Domain and Range
The domain is all real numbers. The range depends on the direction of the parabola.
- Domain: The domain of a quadratic function is always all real numbers (-∞, ∞).
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Range: The range depends on the direction of the parabola:
- If it opens upward (a > 0), the range is [ y-coordinate of vertex, ∞).
- If it opens downward (a < 0), the range is (-∞, y-coordinate of vertex].