Given h(x) = |x + 2| − 3 and ) = −|x| + 4:
a. describe how to obtain the graph of from the graph of a(x) = |x| using transformations.
b. describe how to obtain the graph of h from the graph of a(x) = |x| using transformations.

a)

First translating the graph of a along the x axis towards -2, and then along the y axis towards -3

b)

First we have to reflect the graph along the x axis and then translate it along thw y axis towards 4

Step-by-step explanation:

Hi!

I will consider that stands for g, so we have the fucntions:

h(x) = |x + 2| − 3

g(x) = −|x| + 4

a)

We can see that

h(x) = a(x+2) - 3

this means that we have to do two translations to get from the graph of a(x) to the graph of h(x).

the first one must occur along the x axis, where will the vertex be? we can figure this out looking for the point at which a(x+2) = 0, this point is x=-2, therefore, the first translation will be along the x axis towards -2.

The second translation is along the y axis, and the length of the translation is given evaluating h(x) at the point where a(x+2) vanishes, that is x=-2

h(-2) = -3

Therefore, the second translation will be along the y axis towards -3.

At the end, we will have the same a(x) graph but with its vertex at (-2,-3)

b)

here we can see that:

g(x) = -a(x) + 4

The minus sign multiplying a(x), indicates us that the graph must be reflected along the x axis, that is, insted of ''opening'' upwards, it will ''open'' downwards, the second tranformation is a translation along the y axis, and similar to the previous case, the point will be given when a(x)=0, that is:

g(0) = 4

therefore, the graph must be translated along the y axis towards y = 4

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step-by-step explanation: