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Strike441 [17]
3 years ago
13

Graph y=-2(x+1)^2+4 (must show all work that leads to your graph) please help

Mathematics
1 answer:
bogdanovich [222]3 years ago
3 0

Answer:

See explanation

Step-by-step explanation:

This is a parabola that opens down with the vertex at (-1, 4)  So, find (-1, 4) on your graph.  Now, find y for x = 0 and x = 1 which will give you two points on the parabola.  Since the parabola is symmetrical, find the points that are the partners to the two points you just found.  That should give you a good idea as the graph.

when x = 0,

y = -2(0 + 1)^2 + 4 = -2(1) + 4 = 2    so (0, 2) is on the graph

when x = 1,

y = -2(1 + 1)^2 + 4 = -2(2)^2 + 4 = - 8 + 4 = -4     so (1, -4) is on the graph

Symmetry gives the point (-2, 2) as the partner to (0, 2) and

(-3, -4) as the partner to (1, -4)

I don't have graph  paper to show you that, you are going to have to graph those 5 points ((-1, 4), (0, 2), (1, -4), (-2, 2), (-3, -4)) and draw the parbola.

I hope this helps.  OK?

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Assume {v1, . . . , vn} is a basis of a vector space V , and T : V ------> W is an isomorphism where W is another vector spac
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Answer:

Step-by-step explanation:

To prove that w_1,\dots w_n form a basis for W, we must check that this set is a set of linearly independent vector and it generates the whole space W. We are given that T is an isomorphism. That is, T is injective and surjective. A linear transformation is injective if and only if it maps the zero of the domain vector space to the codomain's zero and that is the only vector that is mapped to 0. Also, a linear transformation is surjective if for every vector w in W there exists v in V such that T(v) =w

Recall that the set w_1,\dots w_n is linearly independent if and only if  the equation

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Recall that w_i = T(v_i) for i=1,...,n. Consider T^{-1} to be the inverse transformation of T. Consider the equation

\lambda_1w_1+\dots \lambda_n w_n=0

If we apply T^{-1} to this equation, then, we get

T^{-1}(\lambda_1w_1+\dots \lambda_n w_n) =T^{-1}(0) = 0

Since T is linear, its inverse is also linear, hence

T^{-1}(\lambda_1w_1+\dots \lambda_n w_n) = \lambda_1T^{-1}(w_1)+\dots +  \lambda_nT^{-1}(w_n)=0

which is equivalent to the equation

\lambda_1v_1+\dots +  \lambda_nv_n =0

Since v_1,\dots,v_n are linearly independt, this implies that \lambda_1=\dots \lambda_n =0, so the set \{w_1, \dots, w_n\} is linearly independent.

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w = a_1w_1+\dots+a_nw_n

Since T is surjective, there exists a vector v in V such that T(v) = w. Since v_1,\dots, v_n is a basis of v, there exist a_1,\dots a_n, such that

a_1v_1+\dots a_nv_n=v

Then, applying T on both sides, we have that

T(a_1v_1+\dots a_nv_n)=a_1T(v_1)+\dots a_n T(v_n) = a_1w_1+\dots a_n w_n= T(v) =w

which proves that w_1,\dots w_n generate the whole space W. Hence, the set \{w_1, \dots, w_n\} is a basis of W.

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