Zeros, x intercepts, passing through x axis or just touching it.

[priceofcarrot]

Homework Statement

Homework Equations

The Attempt at a Solution

So this isn't a specific question, I'm just having trouble with my definitions.

In my book for f(x) = (x - 2) ((x + 1)^2) (x-4) it says that the zeros are at x = 2, -1, 4.

On the page it shows that the graph doesn't cross the x-axis at -1. Does that mean that some zeros cross the x-axis and that some don't? Does -1 count as a zero?

Are all x intercepts zeros? Meaning that x intercepts don't necessarily cross the x-axis either? My book refers to the graph 'bouncing off' at points where the graph doesn't go through a point on the x-axis (where y= 0). So if I'm asked for an x intercept, I can give a value for x where the line only just touches x on the x-axis and doesn't also go through and past x?

 

[Mark44]

So this isn't a specific question, I'm just having trouble with my definitions.

In my book for f(x) = (x - 2) ((x + 1)^2) (x-4) it says that the zeros are at x = 2, -1, 4.

On the page it shows that the graph doesn't cross the x-axis at -1. Does that mean that some zeros cross the x-axis and that some don't? Does -1 count as a zero?

The zeroes don't cross the axis - they are just numbers on the axis. At a zero, a curve can cross or just touch the x-axis. A zero of a function is a number c such that f(c) = 0. Since f(-1) = 0 for your function, then -1 is a zero.

Are all x intercepts zeros? Meaning that x intercepts don't necessarily cross the x-axis either? My book refers to the graph 'bouncing off' at points where the graph doesn't go through a point on the x-axis (where y= 0).

All x-intercepts of a graph occur at the zeroes of the function. X intercepts are just points on the x-axis, so they don't cross it.

So if I'm asked for an x intercept, I can give a value for x where the line only just touches x on the x-axis and doesn't also go through and past x?

Yes.

 

[priceofcarrot]

Thanks for another great reply.