Question

Find the intervals where the graph of the function f(x)= x^2-4 is above the x-axis.

please help?

Answer 1

@doulikepiecauseidont do you understand this question?

Answer 2

find the roots:

x^2-4 = 0 --> x = 2 , -2
between 2, -2 the function is negative (below x-axis) and is +tive outside (above x-axis)

so the solution is:

(-infinity,-2) or ( 2 , infinity)

Answer 3

why be infinity? @SidK

Answer 4

I always do these by drawing a mini graph for visual help. They're not too bad. To find the intervals, it's the best idea to ALWAYS work with the x-intercepts. They will tell you where the graph is positive (above the x-axis) and negative (below the x-axis).

Algebraically, let's find the x-intercepts. Let's turn the equation from standard form into factored form.

Since this equation is a difference of squares, it's really easy to convert into factored form. This easily gives you the x-intercepts.

f(x) = x^2 - 4
f(x) = (x-2)(x+2)

Therefore, the x-intercepts are (2,0) and (-2,0). Refer to graph for a quick visualization. (sorry for the messy drawing but hope it gets the point. Also note parabola opens UP as the coefficient of the first value, a, is positive)


Now looking at the graph, we can see that when the x-values of the graph are LESS THAN -2 and MORE THAN +2, the parabola's line is ABOVE THE X-AXIS. Similarly (not what your q is asking, but for clarification), when the x-values are BETWEEN -2 and +2, your parabola's line is BELOW the x-axis. Take a look, for example, at the x-axis value of 1. The parabola's line at an x-value of 1 is at (1,-3). This is the point the line is in. If you take a look, you can see the parabola's line is BELOW the x-axis. Similarly, if you take the x-value of positive 3, the parabola here is at (3,5). This point of intersection is ABOVE the x-axis. THEREFORE, the intervals of the parabola when the parabola is above x-axis is when x is less than -2 but greater than 2. (-2 > x > 2)

P.S: The intervals do not equal to -2 or 2. This is because at -2 and 2, the interval is neutral as this is the x-axis, and it is neither positive or negative.