# Quadratic function and its graph

A quadratic function is a polynomial function of degree two. It is defined by the formula: \( f(x)=ax^2+bx+c \), where \(a\), \(b\) and \(c\) are constants, and a is not equal to zero.

The graph of a quadratic function is a parabola, which is a U-shaped curve. The direction of the parabola depends on the sign of the leading coefficient \(a\). If \(a\) is positive, the parabola opens upward, and if \(a\) is negative, the parabola opens downward.

The vertex of the parabola is given by the formula: \(\left(-\frac{b}{2a}, \frac{4ac - b^2}{4a}\right)\).

The axis of symmetry of the parabola is the vertical line passing through the vertex, given by the equation \(x=-\frac{b}{2a}\)

The \(x\)-intercepts (zeros) of the quadratic function are given by the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

If the discriminant \(b^2-4ac\) is positive, the quadratic function has two distinct real roots, which are the \(x\)-coordinates of the \(x\)-intercepts. If the discriminant is zero, the quadratic function has one real root, which is the \(x\)-coordinate of the vertex. If the discriminant is negative, the quadratic function has no real roots, but two complex conjugate roots.

Quadratic functions can also be written in factored form: \( f(x)=a(x-r_1 )(x-r_2 ) \), where \(r_1\) and \(r_2\) are the roots of the quadratic function. This form is useful for finding the roots of the function.

The vertex form of the quadratic function is: \( f(x)=a(x-h)^2 +k \), where \( (h,k) \) is the vertex of the parabola. The standard form and vertex form of the quadratic function are related by: \( f(x)=a(x-h)^2 +k=ax^2 -2ahx+ah^2 +k \), which shows that \(a\), \( b=-2ah \), and \( c=ah^2 +k\) are related to the vertex \( (h,k) \).

The quadratic function can be graphed by plotting the vertex, the axis of symmetry, and the \(x\)-intercepts. To sketch the graph, we can also find the maximum or minimum value of the function, the domain, the range, and any transformations or shifts.

**Maximum or minimum value** of the quadratic function:

If \( a > 0 \), the parabola opens upward, and the vertex is the minimum point of the function. The minimum value is \( f(h)=k \). If \( a < 0 \), the parabola opens downward, and the vertex is the maximum point of the function. The maximum value is \(f(h)=k\).

**Domain and range** of the quadratic function:

The domain of the quadratic function is the set of all real numbers, since the function is defined for all values of \(x\). The range depends on the sign of the leading coefficient \(a\). If \(a>0\), the range is \( \left [k, \infty \right ) \) and if \(a < 0 \), the range is \( \left ( - \infty ,k \right ] \).

**Graph:**

The graph of a quadratic function can be transformed by changing the values of \(a\), \(b\) and \(c\). For example, if \(a\) is multiplied by \(a\) positive constant, the graph is stretched vertically, and if \(a\) is multiplied by a negative constant, the graph is reflected about the \(x\)-axis. If \(b\) is added to or subtracted from \(x\), the graph is shifted horizontally and if \(c\) is added to or subtracted from \(f(x)\), the graph is shifted vertically.