# Common Core Standards: Math See All Teacher Resources

#### The Standards

# 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* = 3^{x} will eventually surpass *y* = 3*x* + 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.

x | 3^{x} | 3x + 3 |

1 | 3 | 6 |

2 | 9 | 9 |

3 | 27 | 12 |

4 | 81 | 15 |

5 | 243 | 18 |

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

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 1000^{1001} > 1001^{1000}. Eventually, any exponential function with a base greater than 1 will override polynomial functions.