Determine if the graph of this function is symmetric with respect to the origin. f(x)= 5x^2+6x+9

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Symmetric about the origin also means that if (x,y) is on the graph, then so is (-x,-y).  Here, one point that is on the graph is (1,20):

y = 5(1)^2 + 6(1) + 9 = 20, so (1,20) is on the graph

Then, (-1,-20) would be on the graph.  But:

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Symmetric about the origin also means that if (x,y) is on the graph, then so is (-x,-y).  Here, one point that is on the graph is (1,20):

y = 5(1)^2 + 6(1) + 9 = 20, so (1,20) is on the graph

Then, (-1,-20) would be on the graph.  But:

y = 5(-1)^2 + 6(-1) + 9 = 8, (-1,8) is on the graph.

So, this function isn't symmetric about the origin.

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The graph of the function f(x)= 5x^2+6x+9 is:

For the graph to be symmetric with respect to the origin it should not change when reflected across the x-axis and the y-axis. This is not the case here.

The function f(x)= 5x^2+6x+9 is not symmetric with respect to the origin.

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