Comparing Linear, Quadratic, Polynomial, Exponential Growth - Algebra 2

1. Which grows faster for large $x$: $f(x)=2^x$ or $g(x)=100x^3$?

   $2^x$. Exponential with base $>1$ eventually exceeds any polynomial.

2. Which has constant first differences: linear $mx+b$ or exponential $ab^x$?

   Linear $mx+b$. Linear functions have constant first differences between consecutive outputs.

3. Identify the function type that eventually exceeds all polynomials: linear, polynomial, or exponential.

   Exponential (with base $b>1$). Exponential functions with base $>1$ have fastest long-term growth.

4. Identify whether $f(x)=4x+7$ is linear, polynomial (nonlinear), or exponential.

   Linear. Form $mx+b$ indicates linear function (degree 1).

5. What is the simplest inequality statement for "exponential eventually beats linear" for $b>1$?

   There exists $N$ such that $x>N\Rightarrow b^x>mx+b_0$. Formal statement that exponential eventually dominates linear growth.

6. Identify whether $f(x)=2x^2-3x+1$ is linear, polynomial (nonlinear), or exponential.

   Polynomial (quadratic). Quadratic polynomial has degree 2 (highest power is $x^2$).

7. In a table, if $y$ values are $1,4,9,16$ for $x=1,2,3,4$, what type of function fits exactly?

   Quadratic ($y=x^2$). Values are perfect squares: $1^2, 2^2, 3^2, 4^2$.

8. For $f(x)=ab^x$, what is $f(0)$ in terms of $a$?

   $f(0)=a$. Any number to power 0 equals 1, so $b^0=1$ and $f(0)=a\cdot 1=a$.

9. Which has constant ratio: exponential $ab^x$ or polynomial $x^n$?

   Exponential $ab^x$. Exponential functions have constant ratios between consecutive outputs.

10. Which is larger for sufficiently large $x$: $f(x)=1.2^x$ or $g(x)=0.001x^6$?

    $1.2^x$. Even tiny exponential base $>1$ eventually exceeds high-degree polynomial.

11. In a table, if $y$ values increase by about $7$ each step, what model type is most appropriate?

    Linear growth model. Constant additive increase indicates linear growth pattern.

12. What is the long-run result when comparing $b^x$ with $mx+b_0$ for $b>1$ and $m>0$?

    $b^x$ eventually exceeds $mx+b_0$. Exponential with base $>1$ eventually surpasses any linear function.

13. In a graph, what feature indicates exponential growth rather than linear growth?

    Increasing slope (concave up), not a constant slope. Exponential graphs curve upward with increasing steepness, unlike linear's constant slope.

14. What is the constant additive difference between consecutive outputs of a linear function $f(x)=mx+b$?

    $f(x+1)-f(x)=m$. Linear functions have constant additive differences equal to slope $m$.

15. For $f(x)=2^x$ and $g(x)=x^2$, which is larger at $x=10$?

    $2^{10}$. $2^{10}=1024$ while $10^2=100$, so exponential is much larger.

16. What is the long-run result when comparing $b^x$ with any polynomial $p(x)$ for $b>1$?

    $b^x$ eventually exceeds $p(x)$. Key theorem: exponential with base $>1$ dominates all polynomials eventually.

17. For CCSS.F-LE.3 comparisons, which base range is relevant for "increasing exponentially"?

    $b>1$. Standard compares increasing exponentials, requiring base greater than 1.

18. Which is larger at $x=3$: $f(x)=2^x$ or $g(x)=x^3$?

    $x^3$. $2^3=8$ while $3^3=27$, so polynomial is larger at $x=3$.

19. What is the y-intercept of $f(x)=ab^x$ in terms of $a$ and $b$?

    $(0,a)$. Y-intercept occurs at $x=0$, giving point $(0, a\cdot b^0)=(0,a)$.

20. For $f(x)=2^x$ and $g(x)=3x$, which is larger at $x=10$?

    $2^{10}$. $2^{10}=1024$ while $3(10)=30$, so exponential is much larger.

21. Which grows faster for large $x$: $f(x)=x^5$ or $g(x)=10x^2$?

    $x^5$. Higher degree polynomial grows faster than lower degree polynomial.

22. What limit statement expresses that exponential growth eventually exceeds polynomial growth?

    $\lim_{x\to\infty}\frac{x^n}{b^x}=0$ for $b>1$. Ratio of polynomial to exponential approaches zero as exponential dominates.

23. Which is larger at $x=5$: $f(x)=3^x$ or $g(x)=2x^2$?

    $3^5$. $3^5=243$ while $2(5)^2=50$, so exponential is larger.

24. For $f(x)=5(1.08)^x$, what is the percent increase per $1$ unit of $x$?

    $8\%$. Growth rate is $(1.08-1)\times 100\% = 8\%$.

25. In a table, if $y$ values are $5,9,13,17$ for consecutive $x$, what type of growth is shown?

    Linear (difference $4$). Each value increases by 4 from previous (constant difference).

26. Which grows faster for large $x$: $f(x)=1.01^x$ or $g(x)=x^{10}$?

    $1.01^x$. Even small exponential bases $>1$ eventually dominate high-degree polynomials.

27. What does "eventually exceeds" mean for functions $f$ and $g$ as $x$ increases?

    There exists $N$ such that $x>N\Rightarrow f(x)>g(x)$. Formal definition: after some threshold $N$, $f$ is always greater than $g$.

28. What is the simplest inequality statement for "exponential eventually beats polynomial" for $b>1$?

    There exists $N$ such that $x>N\Rightarrow b^x>p(x)$. General statement that exponential eventually dominates any polynomial.

29. What is the general form of a polynomial function used in comparisons with exponentials?

    $p(x)=a_nx^n+\cdots+a_1x+a_0$ with $a_n\neq 0$. Standard polynomial form with leading coefficient $a_n$ nonzero.

30. What condition on $r$ makes $f(x)=a(1+r)^x$ represent exponential growth?

    $r>0$ (and $a>0$). Growth rate must be positive for $(1+r)^x$ to represent increasing exponential.

31. What is the degree of the polynomial $p(x)=7x^5-2x^3+9$?

    $5$. Degree is the highest power of $x$ in the polynomial.

32. Which grows faster as $x\to\infty$: $f(x)=b^x$ with $b>1$ or $g(x)=x^n$?

    $b^x$ grows faster than $x^n$. Exponential functions with base $b>1$ eventually dominate any polynomial.

33. In a table, if $y$ values are $3,6,12,24$ for consecutive $x$, what type of growth is shown?

    Exponential (ratio $2$). Each value doubles the previous (constant ratio of 2).

34. What condition on $b$ makes $f(x)=ab^x$ an increasing exponential function?

    $b>1$ (with $a>0$). Base must exceed 1 for exponential growth (increasing function).

35. What is the constant multiplicative factor between consecutive outputs of $f(x)=ab^x$?

    $\frac{f(x+1)}{f(x)}=b$. Consecutive outputs have constant multiplicative ratio equal to base $b$.

36. In a table, if $y$ values multiply by about $1.5$ each step, what model type is most appropriate?

    Exponential growth model. Constant multiplicative ratio indicates exponential growth pattern.

37. What is the simplest inequality statement for "exponential eventually beats quadratic" for $b>1$?

    There exists $N$ such that $x>N\Rightarrow b^x>x^2$. Formal statement that exponential eventually dominates quadratic growth.

38. What is the key table test for exponential growth using outputs $y$ at equal $x$-steps?

    Successive ratios $\frac{y_{k+1}}{y_k}$ are constant. Exponential growth shows constant multiplicative ratios in tables.

39. If $0<b<1$ in $f(x)=ab^x$ with $a>0$, what happens to $f(x)$ as $x$ increases?

    $f(x)$ decreases toward $0$. Base $0<b<1$ creates exponential decay toward horizontal asymptote $y=0$.

40. For $f(x)=7\cdot 3^x$, what is $f(0)$?

    $7$. Substitute $x=0$: $f(0)=7\cdot 3^0=7\cdot 1=7$.

41. What is the key table test for linear growth using outputs $y$ at equal $x$-steps?

    Successive differences $y_{k+1}-y_k$ are constant. Linear growth shows constant additive differences in tables.

42. What does it suggest if first differences are not constant but second differences are constant?

    Quadratic growth. Constant second differences indicate quadratic (degree 2) polynomial.

43. Identify the correct comparison for large $x$: $x^4$ vs. $1.0001^x$.

    $1.0001^x$ eventually exceeds $x^4$. Even extremely small exponential base $>1$ eventually exceeds any polynomial.

44. For $f(x)=2^x$ and $g(x)=100x^2$, which is larger at $x=20$?

    $2^{20}$. $2^{20}\approx 1$ million while $100(20)^2=40,000$, exponential dominates.

45. For $f(x)=3\cdot 2^x$, what is the growth factor per $1$ unit increase in $x$?

    $2$. Growth factor is the base $b$ in exponential function $ab^x$.

46. Which grows faster for large $x$: $f(x)=3^x$ or $g(x)=x^2+10x$?

    $3^x$. Exponential with base $>1$ eventually exceeds any polynomial.

47. Which grows faster as $x\to\infty$: $f(x)=e^x$ or $g(x)=x^{100}$?

    $e^x$. Natural exponential eventually dominates any polynomial, regardless of degree.

48. Identify whether $f(x)=7\cdot(1.5)^x$ is linear, polynomial, or exponential.

    Exponential. Form $ab^x$ with constant base indicates exponential function.

49. What is the base $b$ in the exponential function $f(x)=12\cdot(0.7)^x$?

    $0.7$. Base is the number being raised to power $x$ in exponential function.

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