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Common Core Standards: Math

High School: Functions

Linear, Quadratic, and Exponential Models HSF-LE.A.3

3. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

Students should be able to prove that eventually, as long as the functions are headed in the same direction, a quantity increasing exponentially will "beat" linear, quadratic, and polynomial functions. Not much to it.

It's probably obvious that the function y = 3x will eventually surpass y = 3x + 3. We can see this via a table of values or a graph. Somewhere down the line, when x gets closer and closer to infinity, the y value of the exponential function will be larger than the y value of the linear function.

We can see that this happens at x = 2 whether we graph it or look at the table of values.

x3x3x + 3
136
299
32712
48115
524318

What about other functions? Ones with exponents that aren't 1 or x? What about something like y = x1000 compared to y = 1000x? At a large enough x, will 1000x really surpass x1000?

The short answer is that yes, it will. Once x = 1000, the two will be equal. For anything greater, the exponential function will emerge victorious. Because even when x = 1001, we know that 10001001 > 10011000. Eventually, any exponential function with a base greater than 1 will override polynomial functions.

Drills

  1. Which of the following x values proves that f(x) = 2x will surpass the function y = x2 + x + 3?

    Correct Answer:

    40

    Answer Explanation:

    We could either plug these values into the two functions and see which one works (in which case we'd end up with 40), or we could graph the functions and find the point at which 2x surpasses x2 + x + 3.


  2. A local newspaper has projected that its online revenues are growing at a rate modeled by y = 1.5x while its print media only increases at a rate of y = 36x + 35. At which point (in terms of x) do online revenues surpass that of print revenues?

    Correct Answer:

    x < 20

    Answer Explanation:

    You can graph it, solve it mathematically, or construct a table to see that when x is less than 20, our exponential function has surpassed that of the linear one. It took a while, but it got there eventually!


  3. HollerDollar Banking Co. has said that its investment strategy yields a pretty sizeable return rate compared to its competitor bank, Investinus. Its formula for profits models a massive y = x54 + 36x50 + 1500 return rate compared to the other banks, which is modeled by y = 1000x. HollerDollar claims its investment strategy can never be surpassed. Are they correct?

    Correct Answer:

    No, Investinus catches up eventually

    Answer Explanation:

    As x starts out, HollerDollar is definitely in the lead. However, once we start getting into the realm of x = 20, Investinus has caught up sufficiently. By the time we get to x = 27, however, Investinus is somewhere around y = 1 × 1081 while HollerDollar is at 1.97 × 1077, a difference of about 4 orders of magnitude. Maybe we should invest in Investinus.


  4. By simply looking at the equations below, which one will eventually surpass the others?

    Correct Answer:

    y = 100x

    Answer Explanation:

    Because this is an exponential function, it will eventually surpass all other since it increases at an ever-increasing rate. That means linear functions and polynomials are simply no match for it.


  5. Which of the following is a linear equation?

    Correct Answer:

    y = mx + b

    Answer Explanation:

    The standard notation for a linear function is (A). Even though the polynomial answer, (D), can get very large, both the linear and polynomials will be superseded by a (B) or (C) type equation, which are both exponential.


  6. A quadratic function is just a variant of a polynomial function. Is this statement true or false?

    Correct Answer:

    True

    Answer Explanation:

    A quadratic function is essentially written as ax2 + bx + c = y. It's a second-degree polynomial (because of the x2 part). A third-degree polynomial would have x3 instead of x2.


  7. Which of the following is possible if the function y = x + 50 is surpassed by y = 1.1x?

    Correct Answer:

    x < 50

    Answer Explanation:

    The easiest way would be to plug in the values of 10, 20, 40, and 50 into the two functions. While x = 40 will give us 40 + 50 = 90 and 1.140 ≈ 45.3, x = 50 will give us 50 + 50 = 100 and 1.150 ≈ 117.4. That means the right answer is (D).


  8. At what point will the function y = 16x surpass y = 16x?

    Correct Answer:

    x > 1

    Answer Explanation:

    If we substitute x = 1, we get 16 for both (since 161 = 16 × 1). However, anything above instantly makes 16x shoot up into the sky while 16x maintains its relatively slow constant ascent. Even x = 2 makes a substantial difference between 16(2) = 32 and 162 = 256.


  9. Which of the following graphs looks most like an exponential function?

    Correct Answer:

    Answer Explanation:

    Considering what we know about exponential graphs, they need to be constantly increasing. The answer can't be (B) because it increases at a constant rate, not an increasing one. Even though (C) looks promising, we there is no value for x in y = ax that would make y negative. That means that (A) is the right answer.


  10. Given the following graph, which of the lines/curves, most likely represents the exponential function (assuming there's only one)?

    Correct Answer:

    Blue

    Answer Explanation:

    Despite both (A) and (B) looking like curves and having what appear to be ever increasing rates, remember that the exponential function can surpass any polynomial function. That means the exponential function is most likely the one that surpasses all the other ones. So just to be safe, (A) is our best option.


More standards from High School: Functions - Linear, Quadratic, and Exponential Models