Question

the table shows the values of $4^x$ and $100x^2$ for some values of $x$. use the patterns in the table to explain why eventually the values of the exponential expression $4^x$ will overtake the values of the quadratic expression $100x^2$.

Explanation:

Step1: Analyze growth rates

Exponential functions (like \(4^x\)) have exponential growth, where the rate of increase accelerates (multiplicative: \(4^x\) doubles as \(x\) increases by a fixed amount? No, \(4^1 = 4\), \(4^2 = 16\) (multiplied by 4), \(4^3 = 64\) (multiplied by 4), \(4^4 = 256\) (multiplied by 4), \(4^5 = 1024\) (multiplied by 4) – so it grows by a factor of 4 each time \(x\) increases by 1).

Quadratic functions (like \(100x^2\)) have polynomial growth, where the rate of increase is quadratic (additive in the rate: \(100x^2\) at \(x=1\): 100; \(x=2\): 400 (increase by 300); \(x=3\): 900 (increase by 500); \(x=4\): 1600 (increase by 700); \(x=5\): 2500 (increase by 900) – the increase itself increases linearly, but the base growth is \(x^2\)).

Step2: Compare growth over time

• For \(x = 1\) to \(x = 5\):
• \(4^x\) values: 4, 16, 64, 256, 1024 (grows by multiplying by 4 each time).
• \(100x^2\) values: 100, 400, 900, 1600, 2500 (grows by \(100(x+1)^2 - 100x^2 = 100(2x + 1)\), a linear increase in the "jump" between terms).

• At \(x = 5\): \(4^5 = 1024\) and \(100(5)^2 = 2500\) (quadratic is still larger). But exponential growth outpaces polynomial growth in the long run. Mathematically, exponential functions (\(a^x\), \(a > 1\)) grow faster than any polynomial function (\(x^n\), \(n > 0\)) as \(x \to \infty\). This is because the exponent in \(4^x\) causes the function to "explode" in value, while the quadratic \(100x^2\) grows at a slower, polynomial rate.

Step3: Predict long - term behavior

As \(x\) becomes very large (e.g., \(x = 10\)):

• \(4^{10}=1048576\)
• \(100(10)^2 = 10000\)

Here, \(4^{10}\) (exponential) is already much larger than \(100(10)^2\) (quadratic). The pattern in the table shows \(4^x\) growing multiplicatively (factor of 4) and \(100x^2\) growing additively in its "increments" (the difference between consecutive terms increases by 200 each time: 300, 500, 700, 900...). Since exponential growth (multiplicative, accelerating) will eventually surpass polynomial growth (quadratic, which has a fixed degree of growth), \(4^x\) will overtake \(100x^2\) for some large \(x\) (even though it hasn’t yet at \(x = 5\)).

Answer:

Exponential functions (like \(4^x\)) grow by a multiplicative factor (here, ×4 per \(x\)-increase), while quadratic functions (like \(100x^2\)) grow by a polynomial (quadratic) rule. Exponential growth outpaces polynomial growth in the long run: as \(x\) becomes very large, \(4^x\) (exponential) will surpass \(100x^2\) (quadratic) because exponential growth accelerates faster than polynomial growth.