Algebra 2 Flashcards: Comparing Linear Quadratic Polynomial Exponential Growth

What this deck covers

This deck focuses on Comparing Linear Quadratic Polynomial Exponential Growth, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra 2.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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