Struggling with this final question

Question

Hey there!

My name is Cam and I’m struggling with my algebra work right now. My teacher left to have her baby and our new teacher is not good, and doesn’t help or teach us. I really don’t understand quadratics and I feel lost right now.

Here’s the final question, which I don’t understand: https://photos.app.goo.gl/ZiwrZcDW5rjmWy468

I attached the link because I don’t know how to type a table on here. Thank you!

Answer 1

Hello Cam,

This is a very good fundamental question that makes you think about the differences with regards to equation types. You have two different types here, a quadratic and exponential. The easiest way to think about how different they are is relating how they look on a graph... the quadratic creates a parabola, a very smooth bowl shape curve, and the exponential is a line that starts at almost zero to the left and after zero it increases dramatically.

Now obviously, they both increase beyond zero but now the question involves the characteristic of how fast they grow. If you think critically here about the info given to you x = 1,2,3,4 is a very very small subset of input numbers to compare two functions, so pick out larger x values like x = 5,10,15,20. Then compare, you'll find that by the time x = 10 that 2^x is approximately double the result of 5x² . You'll also find that the gap keeps getting larger in favor of the exponential. Therefore, one can conclude that as x increases that an exponential will grow faster than a quadratic. So, Tyler, in the problem, is wrong about his conclusion based on such a small sample size for x, and the fact that he stated a quadratic equation out grows an exponential function.

Answer 2

Cameron,

Try adding a couple more rows to the table and see what happens

x.....5x²....2^x

^{...let's see....}

4....5((4)(4)).....2⁴

4...80....16

5...125...32

6....180...64

....Cameron when I tried the value of x=100.....something magical happens to this table.....it answers the question... what do you think?

Answer 3

Hey Cam....

For both positive and negative values of X, the 5x^2 column always yields a greater value than the 2^x column.

However, when X = 0 the 5x^2 value becomes 5(0)^2=5(0)= 0, where the 2^x value becomes 2^0=1.

Only when X = 0 is the conjecture untrue.

Hope this helps.