Teresa graphs the following 3 equations: y = 2^x, y = x^2 + 2, and y = 2x^2.
She says that the graph of y = 2^x will eventually surpass both of the other graphs.
Is Teresa correct? Why or why not?
Teresa is correct.
The graph of y = 2^x grows at an increasingly increasing rate, but the graphs of y = x^2 + 2 and y = 2x^2 both grow at a constantly increasing rate.
Therefore, the graph of y = 2^x will eventually surpass both of the other graphs.
Teresa is not correct.
The graph of y = 2^x grows at an increasing rate and will eventually surpass the graph of y = x^2 + 2.
However, it will never surpass the graph of y = 2x^2 because the y-value is always twice the value of x^2.
Teresa is not correct.
The graph of y = 2x^2 already intersected and surpassed the graph of y = 2^x at x = 1.
Once a graph has surpassed another graph, the other graph will never be higher.
Teresa is not correct.
The graph of ( y = 2^x ) does grow exponentially, while ( y = 2x^2 ) grows quadratically. However, for values of ( x ) greater than 0, ( 2^x ) will eventually surpass both ( y = x^2 + 2 ) and ( y = 2x^2 ). This is because exponential functions outpace quadratic functions in growth as ( x ) increases.
Therefore, while it may seem that ( y = 2x^2 ) could be higher at lower values of ( x ), as ( x ) progresses, ( y = 2^x ) will indeed surpass both of the other graphs eventually.