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Answer:
For a function f(x), the y-intercept is the value of f(0), while the x-intercepts are the values of x such that:
f(x) = 0.
Here we can also start with some information about dilations.
For a function f(x), a vertical dilation of scale factor k is written as:
g(x) = k*f(x)
While, again, for a function f(x), an horizontal dilation of scale factor k is written as:
g(x) = f(x/k)
So when here they ask us the difference between f(2x) and 2f(x)
We can say that the first one is a horizontal dilation of scale factor 1/2, while the second is a vertical dilation of scale factor 2.
Now let's analyze the intercepts for each case.
if f(x) = |x + 2| - 3
2*f(x) = 2*(|x + 2| - 3)
= 2*|x + 2| - 6
The y-intercept is:
2*f(0) = 2*|0 + 2| - 6 = 4 - 6 = -2
and the x-intercepts are given by:
2*f(x) = 0 = 2*|x + 2| - 6
6 = 2*|x + 2|
6/2 = |x + 2|
3 = |x + 2|
Here we have two solutions for x,
x = 1: 3 = |1 + 2| = |3| = 3
x = -5: 3 = |-5 + 2| = |-3| = 3
Now for the other transformation, we have:
f(2*x) = |2*x + 2| - 3
the y-intercept is:
f(2*0) = |2*0 + 2| - 3
= |2| - 3 = -1
(So the y-intercept is different)
And the x-intercepts are given by:
f(2*x) = 0 = |2*x + 2| - 3
3 = |2*x + 2|
3 = 2*|x + 1|
3/2 = |x + 1|
Here the two solutions are:
x = 1/2
x = -5/2
So the x-intercepts are also differents.
This happens because in one case, we affect the function horizontally and in the other case vertically, so one could expect that the resulting functions are different (thus, have different intercepts)