Help with quadratic function reasoning

[tommyjefferson7]

Hello, I'm studying and I can't figure out this one:

Consider a quadratic equation written in vertex form, f(x) = a(x − h)2 + k

1. Explain in sentences why the graph of this function will have no x-intercepts (also called roots) if a and k have like signs

Thank you!

 

[Dr.Peterson]

Suppose first that a and k are both positive. Where will the vertex be? Which direction will the parabola open? Can it cross the x-axis by going in that direction?

Then try the other case, where they are both negative.

 

[Steven G]

I'll try a slightly different approach than Dr Peterson.

Assume a and k are both positive: What can you say about the sign of (x-h)^2? How about the sign of a(x-h)^2? How about the sign of a(x-h)^2?

What can you say if a and k are both negative?