1. Domain :
The domain of a quadratic function f(x) = ax^{2 }+ 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) = ax^{2 }+ bx + c are the two values of x when f(x) = 0 or ax^{2} + bx + c = 0.
6. The zeros of a quadratic function f(x) = ax^{2 }+ 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) = ax^{2 }+ bx + c is -b/a.
14. The product of the zeros of the quadratic function f(x) = ax^{2}+ 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) = ax^{2} + bx + c, if the sign of the first term (ax^{2}) is negative, the parabola will be open downward. Otherwise, the parabola will be open downward.
24. The discriminant b^{2} - 4ac discriminates the nature of the zeros of the quadratic function f(x) = ax^{2} + bx + c.
Let us see how this discriminant b^{2} - 4ac can be used to know the nature of the roots of a quadratic function.
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