Mathematics QUADRATIC EXPRESSIONS AND THEIR GRAPHS

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QUADRATIC EXPRESSIONS AND THEIR GRAPHS

Quadratic Expression And Its Graphs :

In `y = ax^2 + bx + c` , if `a, b, c in R` and `a ne 0` . Graph of quadratic takes the shape of a parabola. The parabola opens upward or downward according as `a> 0` or `a< 0` respectively.

The lowest point `P` in figure-`(i)` and highest point `Q` in figure-`(ii)` is called as vertex of parabola. Now for different values of `a, b, c` if graph `y = ax^2 + bx +c` is plotted then following `6` different shapes are obtained.

Case-I : If `a > 0` and `D > 0`

Then quatratic equation has two roots and graph cuts the `x`-axis at two distinct points.

(i) For `alpha < x < beta => y` is negative.

(ii) For `x < alpha` or `x > beta => y` is positive.

Case-ll: If `a > 0` and `D = 0`

Then curve touches `x`-axis. Hence both zeroes of polynomial coincides.

In this type equation becomes `y = a(x - a.)^2` and `y ge 0`, for `x in R`.

Case-llI: If `a > 0` and `D < 0`

Then curve completely lies above `x`-axis.

In this case imaginary roots appears and `y > 0` for `x in R` .

Case-IV : If `a < 0` and `D > 0`

Then graph is downward and cuts the `x`-axis at two distinct points.

In this case (a) `y > 0` if `alpha < x < beta`

(b) `y < 0` , if `x < alpha` or `x > beta`

Case-V : If `a < 0` and `D = 0`

Then graph touches the `x`-axis from below.

In this case `x in R, y le 0` for `x in R`

Case-VI : If `a < 0` and `D < 0`

Then graph lies completely below the `x`-axis and `y<0` for `x in R` .