Prove algebraically that y=x^2-4 is symmetric with the y-axis.
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let f(x) = x^2 - 4
f(a) = a^2 - 4
f(-a) = a^2 - 4
if for some function f(x),
f(a) = f(-a) then the function is symmetric about the y-axis
(by definition)
To prove algebraically that the graph of the equation y = x^2 - 4 is symmetric with the y-axis, we need to show that replacing x with -x in the equation also gives us a valid equation. If the equation remains the same after this substitution, it indicates that the graph is symmetric with respect to the y-axis.
Let's substitute -x for x in the equation y = x^2 - 4:
y = (-x)^2 - 4
y = x^2 - 4
As we can see, the equation remains the same after replacing x with -x. This reaffirms that the graph is symmetric with respect to the y-axis.
In more detail, replacing x with -x yields the same equation, which means both sides of the equation will have the same value for any given value of x. Therefore, the graph of y = x^2 - 4 will be symmetrical along the y-axis, with any corresponding points on opposite sides of the y-axis having the same y-coordinate.
This can also be observed visually by graphing the equation or by plotting a table of values. If the graph is symmetrical with respect to the y-axis, it will appear the same on both sides of the y-axis.