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A quadratic function is a polynomial function with one or more variables in which the highest exponent of the variable is two. Since the highest degree term in a quadratic function is of the second degree, therefore it is also called the polynomial of degree 2. A quadratic function has a minimum of one term which is of the second degree. It is an algebraic function.

A quadratic function is one of the form f(x) = ax2 + bx + c, where a, b, and c are numbers with a not equal to zero. The graph of a quadratic function is a curve called a Parabola. Parabolas may open upward or downward and vary in "width" or "steepness", but they all have the same basic "U" shape.

**Three properties that are universal to all quadratic functions:**

- The graph of a quadratic function is always a parabola that either opens upward or downward (end behavior);
- The domain of a quadratic function is all real numbers; and
- The vertex is the lowest point when the parabola opens upwards; while the vertex is the highest point when the parabola opens downward.
- The standard form of the quadratic function is f(x) = ax
^{2}+bx+c where a ≠ 0. - The graph of the quadratic function is in the form of a parabola.
- The quadratic formula is used to solve a quadratic equation ax
^{2}+ bx + c = 0 and is given by x = −b±√b2−4ac/2a - The discriminant of a quadratic equation ax
^{2}+ bx + c = 0 is given by b^{2}-4ac. This is used to determine the nature of the solutions of a quadratic function.

A quadratic function can always be factorized, but the factorization process may be difficult if the zeroes of the expression are non-integer real numbers, or non-real numbers. In such cases, we can use the quadratic formula to determine the zeroes of the expression. The general form of a quadratic function is given as: f(x) = ax^{2} + bx + c, where a, b, and c are real numbers with a ≠ 0. The roots of the quadratic function f(x) can be calculated using the formula of the quadratic function which is:

x=−−b±√b2−4ac/2a

**The quadratic function equation is f(x) = ax ^{2} + bx + c, where a ≠ 0. Let us see a few examples of quadratic functions:**

- f(x) = 2x
^{2}+ 4x - 5; Here a = 2, b = 4, c = -5 - f(x) = 3x
^{2}- 9; Here a = 3, b = 0, c = -9 - f(x) = x
^{2}- x; Here a = 1, b = -1, c = 0

Now, consider f(x) = 4x-11; Here a = 0, therefore f(x) is not a quadratic function.

*Example 1*

Sketch the graph of y = x2/2. Starting with the graph of y = x^{2}, we shrink by a factor of one-half. This means that for each point on the graph of y = x^{2}, we draw a new point that is one-half of the way from the x-axis to that point.

*Example 2*

Sketch the graph of y = (x - 4) ^2 - 5. We start with the graph of y = x^{2}, shift 4 units right, then 5 units down.

*Example 3: Alternate method of finding the vertex*

In some cases, completing the square is not the easiest way to find the vertex of a parabola. If the graph of a quadratic function has two x-intercepts, then the line of symmetry is the vertical line through the midpoint of the x-intercepts.

The x-intercepts of the graph above are at -5 and 3. The line of symmetry goes through -1, which is the average of -5 and 3. (-5 + 3)/2 = -2/2 = -1. Once we know that the line of symmetry is x = -1, then we know the first coordinate of the vertex is -1. The second coordinate of the vertex can be found by evaluating the function at x = -1.

A quadratic function is one of the form f(x) = ax2 + bx + c, where a, b, and c are numbered with a not equal to zero. The graph of a quadratic function is a curve called a parabola. Parabolas may open upward or downward and vary in "width" or "steepness", but they all have the same basic "U" shape.

**There are three commonly-used forms of quadratics:**

- Standard Form: y = ax
^{2}+bx+c*y* - Factored Form: y = a(x-r
_{1}) (x-r_{2}) - Vertex Form: y = a (x-h)
^{2}+k

**There are several methods you can use to solve a quadratic equation:**

- Factoring
- Completing the Square
- Quadratic Formula
- Graphing

: f(x) = ax*Standard form*^{2}+ bx + c, where a ≠ 0.f(x) = a (x - h)*Vertex form:*^{2}+ k, where a ≠ 0 and (h, k) is the vertex of the parabola representing the quadratic function.: f(x) = a (x - p) (x - q), where a ≠ 0 and (p, 0) and (q, 0) are the x-intercepts of the parabola representing the quadratic function.*Intercept form*

A quadratic is an expression of the form ax^{2} + bx + c, where a, b and c are given numbers and a ≠ 0. The standard form of a quadratic equation is an equation of the form

ax^{2} + bx + c = 0, where a, b and c are given numbers and a ≠ 0.

A non-monic quadratic equation is an equation of the form ax^{2} + bx + c = 0, where and are given numbers, and a ≠ 1 or 0. This is the general case. Thus 2x^{2} + 5x + 3 = 0 is an example of a non-monic quadratic equation.

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