Comparing Linear, Quadratic, Polynomial, Exponential Growth - Algebra 1

1. Which increases by multiplication: $f(x)=8\cdot(1.3)^x$ or $g(x)=8x+1.3$?

   $f(x)=8\cdot(1.3)^x$. Exponential form multiplies by base; linear form adds slope.

2. What does "eventually exceeds" mean when comparing $f(x)$ and $g(x)$?

   There exists $N$ such that for all $x>N$, $f(x)>g(x)$. Eventually exceeds means one function surpasses another for large $x$.

3. Which eventually becomes larger as $x\to\infty$: $0.5x^2$ or $3^x$?

   $3^x$ eventually becomes larger than $0.5x^2$. Exponential functions eventually outgrow quadratic functions.

4. What is the $y$-intercept of $f(x)=mx+b$?

   $b$. Linear functions equal their constant term $b$ when $x=0$.

5. Which grows faster for large $x$: $x^4-x$ or $1.2^x$?

   $1.2^x$ eventually grows faster than $x^4-x$. Exponential growth exceeds high-degree polynomials for large $x$.

6. What is the standard form of a polynomial function used for growth comparisons?

   $p(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0$ with $n\ge 1$. Standard polynomial form with degree $n$ and leading coefficient $a_n$.

7. In a table, which pattern indicates polynomial (nonlinear) rather than exponential growth?

   Differences change, and ratios are not constant. Polynomial functions have varying differences and ratios between terms.

8. Which eventually becomes larger as $x\to\infty$: $100x^{10}$ or $1.05^x$?

   $1.05^x$ eventually becomes larger than $100x^{10}$. Exponential dominates polynomial regardless of coefficients or degree.

9. Which is the best general comparison: exponential vs. polynomial as $x\to\infty$?

   Exponential growth dominates polynomial growth for large $x$. This summarizes the fundamental exponential vs polynomial comparison.

10. Which grows faster for large $x$: $10x$ or $1.001^x$?

    $1.001^x$ eventually grows faster than $10x$. Even tiny exponential bases eventually surpass linear growth.

11. What is the common ratio for $f(x)=12\cdot 0.9^x$ in a table with step size $1$?

    $0.9$. Base $0.9$ means each step multiplies by $0.9$.

12. If $f(x)=3\cdot(1.5)^x$ and $g(x)=100x^3$, what is true for large $x$?

    $f(x)>g(x)$ for sufficiently large $x$. Exponential functions eventually dominate polynomial functions.

13. Which is exponential: $f(x)=5x^3$ or $g(x)=5\cdot 3^x$?

    $g(x)=5\cdot 3^x$. Form $a \cdot b^x$ with variable in exponent indicates exponential.

14. Which has constant ratio in a table: $f(x)=7x+1$ or $g(x)=3\cdot 2^x$?

    $g(x)=3\cdot 2^x$ has constant ratio $2$. Exponential functions have constant ratios, not linear functions.

15. What is the $y$-intercept of $f(x)=a\cdot b^x$?

    $f(0)=a$. Exponential functions equal their coefficient $a$ when $x=0$.

16. What is the degree of the polynomial $p(x)=7x^5-3x^2+1$ used in comparisons?

    $5$. The highest power of $x$ determines polynomial degree.

17. For $f(x)=mx+b$, what is $f(x+1)-f(x)$?

    $m$. The slope $m$ is the constant difference for linear functions.

18. Which is polynomial: $f(x)=4^x+2$ or $g(x)=4x^2+2$?

    $g(x)=4x^2+2$. Variable raised to constant power indicates polynomial function.

19. If $f(2)=12$ and $f(3)=18$ for an exponential model, what is $\frac{f(3)}{f(2)}$?

    $1.5$. For exponential functions, consecutive ratios equal the base.

20. If $f(0)=10$ and $f(1)=15$, what is $b$ in $f(x)=a\cdot b^x$?

    $b=1.5$. The base $b$ equals the ratio $\frac{f(1)}{f(0)}=\frac{15}{10}=1.5$.

21. If $f(x)=2^x$ and $g(x)=x^2$, what is true about $f(x)-g(x)$ for large $x$?

    It becomes positive and keeps increasing. Exponential $f(x)$ eventually exceeds polynomial $g(x)$, so difference grows.

22. What does it mean if a graph becomes steeper and steeper as $x$ increases?

    It suggests exponential (or faster-than-linear) growth. Increasing steepness indicates accelerating growth like exponential functions.

23. Choose which will be larger for sufficiently large $x$: $f(x)=x^7$ or $g(x)=1.01^x$.

    $g(x)=1.01^x$. Even small exponential base dominates high-degree polynomial.

24. For $f(x)=a\cdot b^x$ with $a>0$, what is $\frac{f(x+1)}{f(x)}$?

    $b$. The base $b$ is the constant ratio for exponential functions.

25. Which grows faster for large $x$: $x^2+1000$ or $1.02^x$?

    $1.02^x$ eventually grows faster than $x^2+1000$. Exponential growth dominates quadratic even with large constants.

26. If a table shows $f(0)=5$, $f(1)=9$, $f(2)=13$, what is the growth type?

    Linear growth (difference $4$). Adding $4$ each step indicates linear growth with slope $4$.

27. What is the defining feature of exponential growth visible in a table of values?

    A constant ratio $\frac{f(x+1)}{f(x)}=b$ (approximately). Exponential functions have constant ratios between consecutive terms.

28. What is the common difference for $f(x)= -3x+10$ in a table with step size $1$?

    $-3$. Slope $-3$ means each step subtracts $3$.

29. Which grows faster for large $x$: $2x^2$ or $1.1^x$?

    $1.1^x$ eventually grows faster than $2x^2$. Exponential functions eventually outgrow all polynomial functions.

30. Which term determines the end behavior of a polynomial $p(x)=a_nx^n+\cdots$ for large $x$?

    The leading term $a_nx^n$. The highest degree term dominates behavior for large $x$ values.

31. Identify the growth type of $f(x)=6\cdot(0.8)^x$.

    Exponential decay. Base $0.8 < 1$ indicates exponential decay pattern.

32. If a table shows $f(0)=3$, $f(1)=6$, $f(2)=12$, what is the growth type?

    Exponential growth (ratio $2$). Doubling pattern indicates exponential growth with base $2$.

33. Which grows faster for large $x$: $x^5$ or $1.0001^x$?

    $1.0001^x$ eventually grows faster than $x^5$. Exponential growth dominates polynomial growth regardless of degree.

34. Which condition on $b$ gives exponential decay for $f(x)=a\cdot b^x$ with $a>0$?

    $0<b<1$. Base between $0$ and $1$ causes exponential decay.

35. What does it mean for a function to show exponential growth in the form $f(x)=a\cdot b^x$?

    $b>1$ (with $a>0$) so outputs multiply by the same factor each step. Exponential growth multiplies by a constant factor $b$ at each step.

36. Choose which will be larger for sufficiently large $x$: $f(x)=1000x$ or $g(x)=1.1^x$.

    $g(x)=1.1^x$. Exponential growth with base $1.1$ eventually dominates linear growth.

37. Which has constant first differences: $f(x)=3x^2$ or $g(x)=2^x$?

    Neither; constant first differences occur for linear functions. Only linear functions have constant first differences in tables.

38. Choose which will be larger for sufficiently large $x$: $f(x)=x^2$ or $g(x)=2^x$.

    $g(x)=2^x$. Exponential base $2$ eventually exceeds quadratic polynomial growth.

39. Which grows faster for large $x$: $5x+7$ or $1.01^x$?

    $1.01^x$ eventually grows faster than $5x+7$. Even small exponential bases like $1.01$ eventually dominate linear functions.

40. If $f(0)=2$, $f(1)=4$, $f(2)=8$, what is an explicit rule $f(x)=a\cdot b^x$?

    $f(x)=2\cdot 2^x$. Initial value $a=2$ and base $b=2$ from the doubling pattern.

41. What is the defining feature of linear growth visible in a table of values?

    A constant difference $f(x+1)-f(x)=m$. Linear functions add the same amount $m$ between consecutive terms.

42. What is the key CCSS.F-LE.3 comparison statement between exponential and polynomial growth?

    Exponential growth eventually exceeds any polynomial growth. This is the core principle of CCSS.F-LE.3 about exponential dominance.

43. What is one reliable table-based method to decide if data are exponential rather than linear?

    Check for a constant ratio, not a constant difference. Constant ratios distinguish exponential from linear growth patterns.

44. Which eventually becomes larger as $x\to\infty$: $x^3$ or $2^x$?

    $2^x$ eventually becomes larger than $x^3$. Exponential functions eventually dominate polynomial functions.

45. What does it mean for a function to show linear growth in the form $f(x)=mx+b$?

    It increases by a constant add-on amount for equal $x$-steps. Linear functions have constant slope $m$, adding the same amount each step.

46. What is the key shape difference on a graph between exponential and linear growth?

    Exponential curves upward more and more; linear is a straight line. Exponential curves get steeper while linear graphs maintain constant slope.

47. Identify the growth type of $f(x)=6\cdot(1.2)^x$.

    Exponential growth. Base $1.2 > 1$ indicates exponential growth pattern.

48. What is the growth type of $f(x)=x^4-2x+9$?

    Polynomial growth (degree $4$). Highest power term determines polynomial degree and growth rate.

49. Which condition on $b$ gives exponential growth for $f(x)=a\cdot b^x$ with $a>0$?

    $b>1$. Base greater than $1$ causes exponential growth.

50. If $f(0)=7$ and the common ratio is $3$, what is $f(2)$ for $f(x)=a\cdot b^x$?

    $63$. With $a=7$ and $b=3$, we get $f(2)=7 \cdot 3^2=63$.

51. What is the key shape difference on a graph between exponential and quadratic growth?

    Exponential eventually becomes steeper than any parabola. Exponential growth rate increases while quadratic growth rate is bounded.

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