PROPERTIES OF QUADRATIC FUNCTIONS

1. Domain : 

The domain of a quadratic function f(x) = ax² + bx + c is all real numbers. The graph of a quadratic function is a parabola. 

2. The vertex form of a quadratic function is 

f(x) = a(x - h)² + k

where (h, k) is the vertex. 

3.  In f(x) = a(x - h)² + k, if a > 0, the  parabola opens up and if a < o, the parabola opens down.

4. Range : 

In the vertex form of (x) = a(x - h)² + k,

(i) if a > 0 (parabola opens up), the range is [k,∞).

(ii) if a < 0 (parabola opens down), the range is (-∞, k].

5. The zeros of a quadratic function f(x) = ax² + bx + c are the two values of x when f(x) = 0 or ax² + bx + c = 0.

6. The zeros of a quadratic function f(x) = ax² + bx + c are the two x-intercepts of the parabola. 

7. The number of x-intercepts of a quadratic function depends on whether the graph opens up or down and it also depends on whether the vertex is above or below the x-axis.

8. If the graph of a quadratic function opens up and the vertex is above the x-axis or if the graph opens down and the vertex is below the x-axis, then there will be no x-intercepts.

9. If the vertex is touching the x-axis, then there is one x-intercept regardless of whether the graph opens up or down.

10. If the graph of a quadratic function opens up and the vertex is below the x-axis or if the graph opens down and the vertex is above the x-axis, then there will be two x-intercepts.

11. There are three methods to find the two zeros (x-intercepts) of a quadratic function. They are, 

(i) Factoring

(ii) Quadratic formula

(iii) Completing square

12. If the two zeros of a quadratic function are irrational, then the two zeros (roots) will occur in conjugate pairs. That is, if (m + √n)is a root, then (m - √n) is the other root of the same equation. 

13. The sum of the zeros of the quadratic function f(x) = ax² + bx + c is -b/a.

14. The product of the zeros of the quadratic function f(x) = ax²+ bx + c is c/a.

15. If one zero is reciprocal to the other root then their product c/a = 1 or c = a. 

16. If one root is equal to other root but opposite in sign then their sum = 0. That is, b/a = 0, so b = 0. 

17. The graph of any quadratic function will be a parabola.

18. The zeros of a quadratic equation are the x-coordinates of the points where the parabola (graph of quadratic a function) cuts x-axis.

19. If the two zeros of a quadratic function are imaginary, then the graph (parabola) will never intersect x - axis. 

20. The two x-intercepts of a parabola (graph of a quadratic function) are nothing but the zeros of the quadratic function. 

21. x- coordinate of the vertex of the parabola is -b/2a and the vertex is (-b/2a, f(-b/2a)). 

22. To know at where the parabola cuts y-axis or y-intercept of the parabola, we have to plug x = 0 in the given quadratic function.

23. f(x) = ax² + bx + c, if the sign of the first term (ax²) is negative, the parabola will be open downward. Otherwise, the parabola will be open downward. 

24. The discriminant b² - 4ac  discriminates the nature of the zeros of the quadratic function f(x)  =  ax² + bx + c.

Let us see how this discriminant  b² - 4ac can be used to know the nature of the roots of a quadratic function. 

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