Question

for each of the following, state the condition on a and k such that the parabola y=a(x-h)^2+k has the given property.

a) The parabola intersects the x-axis at two distinct points
b)The parabola intersects the x-axis at one point.
c)The parabola does not intersect the x-axis.

Answer 1

Consider the parabola $y=a(x-h)^2+k$

a) The parabola intersect the $x$ -axis on setting $y=0$

So,

$a(x-h)^2+k=0$

$a(x-h)^2=-k$

$(x-h)^2=\frac{-k}{a}$

$x=h\pm\sqrt{-\frac{k}{a}}$

For real values of $x$ , the term under square root sign must be positive.

Therefore, the condition for two distinct points are:

i) For $a>0$ , the value of $k$ should be negative, that is $k<0$

ii) For $k>0$, the value of $a$ should be negative, that is $a<0$

b) For $k=0$ , the $x$ -intercept is,

$x=h\pm\sqrt{-\frac{0}{a}}$

$=h\pm0$

$=h$

Therefore, the parabola intersect the $x$ -axis at one point when $k=0$ irrespective of the value of $a$ .

c) Here,

$x=h\pm\sqrt{-\frac{k}{a}}$

The $x$ value will be undefined if $\frac{k}{a}>0$

Therefore, the condition for the parabola not intersecting the $x$ -axis are:

i) For $a>0$ , the value of $k$ should be positive, that is $k>0$

ii) For $a<0$, the value of $k$ should be negative, that is $k<0$