The question I've been trying to figure out is asking me to explain how the graph of (g) is obtained from the graph of (f).
The graphs are as followed:
- f(x)= x^(2)-4
- g(x)= [x^(2)-4]
(Note: the Brackets represent absolute value symbols around the expression for g(x))
take a look at the graphs and see what you can say.
And why not use || for absolute value? It's not like you don't have the characters...
http://www.wolframalpha.com/input/?i=plot+y%3Dx^2-4,+y%3D|x^2-4|
I looked at the graph and they both look similar. I'm not sure how to put it into the proper words.
Is there anyway you could explain it to me? I'm having trouble.
PLEASE
come on. x^2-4 is negative on the interval (-2,2)
So, since |x^2-4| is always positive, that little arc below the x-axis is flipped up above it.
Everywhere else, the two functions are identical.
g(x) = -f(x) if |x| < 2
g(x) = f(x) if |x| >= 2
To understand how the graph of g(x) is obtained from the graph of f(x), we need to first analyze the functions f(x) = x^2 - 4 and g(x) = |x^2 - 4|.
The function f(x) = x^2 - 4 represents a parabola that opens upwards and is shifted downward by 4 units compared to the parent function y = x^2. This means that the vertex of the parabola is at (0, -4), and the graph extends infinitely in both the positive and negative x-directions.
On the other hand, the function g(x) = |x^2 - 4| represents the absolute value of the expression x^2 - 4. The effect of taking the absolute value is that the negative values of the expression become positive. In the context of the graph, it means that any portion of the graph that lies below the x-axis will be reflected and shifted above the x-axis.
To obtain the graph of g(x) from the graph of f(x), we need to consider the points where f(x) crosses or lies below the x-axis. These are the points where the function g(x) will be affected.
In the case of f(x) = x^2 - 4, the function crosses the x-axis at x = -2 and x = 2. At these x-values, f(x) is equal to zero. Plugging these values into g(x) = |x^2 - 4|, we find g(-2) = |-2^2 - 4| = 0 and g(2) = |2^2 - 4| = 0.
Therefore, on the graph of g(x), we include the points (2, 0) and (-2, 0) where f(x) crosses the x-axis.
Additionally, for any portion of the graph of f(x) that lies below the x-axis, we reflect it above the x-axis to obtain the graph of g(x). In this case, the portion below the x-axis is the part of the parabola between x = -2 and x = 2. We flip this portion above the x-axis to create a symmetric graph for g(x).
Overall, the graph of g(x) will have the same shape as the graph of f(x) except that any points where f(x) crosses or lies below the x-axis will be mirrored and shifted above the x-axis.