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Given equation is [tex]f\left(x\right)=x^2-b^2[/tex].

Now we need to find about what are the key aspects of the graph of [tex]f\left(x\right)=x^2-b^2[/tex], where b is a real number.

We know that square of any number is always positive.

then [tex]b^2[/tex] must be a positive number.

So that means for any real number b, as the value of b increases then graph of f(x) shifts downward by [tex]b^2[/tex] units as compared to the graph of parent function [tex]f\left(x\right)=x^2[/tex]

Answer with explanation:

The graph of the function is:

   f(x)=x² -b²

Here, b is any Real Number.

f(x)=x² - k, where, k=b².

→y+k=x²

The given curve represents a Parabola having vertex at ,(0, -k) which can be Obtained by , putting, x=0 and, y+k=0→y= -k.

→The curve will open vertically Upwards having y axis as Line of Axis.

→It will cut, x axis at two points, if , k<0 and does not cuts the x axis , if k>0.

→Line, x=0, divides the Parabola into two equal Parts.

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