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: 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 2: 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 3: 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 4: 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 5: 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 6: 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 7: 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 8: 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 9: 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 10: 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 11: 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 12: Which has constant ratio: exponential $ab^x$ or polynomial $x^n$?

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

Flashcard 13: 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 14: 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 15: 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 16: Identify whether $f(x)=4x+7$ is linear, polynomial (nonlinear), or exponential.

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

Flashcard 17: 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 18: 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 19: 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 20: 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 21: 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 22: 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 23: 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 24: 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 25: 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 26: 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 27: 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 28: 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 29: 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 30: For $f(x)=7\cdot 3^x$, what is $f(0)$?

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