If a and h are positive constants and k is a negative constant, how many times will the graph of y = a(x – h)2 + k intersect the x-axis?

Question

 Help! I'm stuck.

Answer 1

 

Hi Adriana,

 

 

y = a(x-h)² + k represents a parabola of which vertex is (h,k).

 

A parabola can intersect the x-axis up to twice

 

depending on a and k in the equation above.

 

1. If a and k have a different polarity,

 

   (that is, a is positive and k is negative,

 

    or the other way around)

 

    a parabola intersects the x-axis twice.

 

2. If k = 0, a parabola meets with the x-axis only at one point

 

    regardless of whether a is positive or negative.

 

3. If a and k have the same polarity,

 

    a parabola never meets with the x-axis.

 

 

In this problem where a is positive,

 

a parabola is open upwards.

 

And as h is positive and k is negative,

 

its vertex is in the fourth quadrant.

 

So, the given parabola meets with the x-axis twice.

 

Hope this helps.