Algebra 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.

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

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

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

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

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

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

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

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

Flashcard 5: Which grows faster for large $x$: $10x$ or $1.001^x$?

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

Flashcard 6: Which grows faster for large $x$: $x^4-x$ or $1.2^x$?

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

Flashcard 7: What is the standard form of a polynomial function used for growth comparisons?

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

Flashcard 8: If $f(x)=3\cdot(1.5)^x$ and $g(x)=100x^3$, what is true for large $x$?

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

Flashcard 9: In a table, which pattern indicates polynomial (nonlinear) rather than exponential growth?

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

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

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

Flashcard 11: Choose which will be larger for sufficiently large $x$: $f(x)=x^7$ or $g(x)=1.01^x$.

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

Flashcard 12: Which is the best general comparison: exponential vs. polynomial as $x\to\infty$?

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

Flashcard 13: Which grows faster for large $x$: $10x$ or $1.001^x$?

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

Flashcard 14: Which eventually becomes larger as $x\to\infty$: $100x^{10}$ or $1.05^x$?

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

Flashcard 15: What is the common ratio for $f(x)=12\cdot 0.9^x$ in a table with step size $1$?

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

Flashcard 16: If $f(x)=3\cdot(1.5)^x$ and $g(x)=100x^3$, what is true for large $x$?

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

Flashcard 17: Which is exponential: $f(x)=5x^3$ or $g(x)=5\cdot 3^x$?

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

Flashcard 18: Which has constant ratio in a table: $f(x)=7x+1$ or $g(x)=3\cdot 2^x$?

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

Flashcard 19: What is the $y$-intercept of $f(x)=a\cdot b^x$?

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

Flashcard 20: What is the degree of the polynomial $p(x)=7x^5-3x^2+1$ used in comparisons?

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

Flashcard 21: If $f(0)=10$ and $f(1)=15$, what is $b$ in $f(x)=a\cdot b^x$?

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

Flashcard 22: For $f(x)=mx+b$, what is $f(x+1)-f(x)$?

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

Flashcard 23: Which is polynomial: $f(x)=4^x+2$ or $g(x)=4x^2+2$?

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

Flashcard 24: What does "eventually exceeds" mean when comparing $f(x)$ and $g(x)$?

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

Flashcard 25: What is the $y$-intercept of $f(x)=mx+b$?

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

Flashcard 26: If $f(2)=12$ and $f(3)=18$ for an exponential model, what is $\frac{f(3)}{f(2)}$?

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

Flashcard 27: If $f(0)=10$ and $f(1)=15$, what is $b$ in $f(x)=a\cdot b^x$?

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

Flashcard 28: If $f(x)=2^x$ and $g(x)=x^2$, what is true about $f(x)-g(x)$ for large $x$?

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

Flashcard 29: What does it mean if a graph becomes steeper and steeper as $x$ increases?

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

Flashcard 30: What is the common difference for $f(x)= -3x+10$ in a table with step size $1$?

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

Flashcard 31: What is the growth type of $f(x)=x^4-2x+9$?

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

Flashcard 32: Choose which will be larger for sufficiently large $x$: $f(x)=x^7$ or $g(x)=1.01^x$.

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

Flashcard 33: For $f(x)=a\cdot b^x$ with $a>0$, what is $\frac{f(x+1)}{f(x)}$?

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

Flashcard 34: Which grows faster for large $x$: $x^2+1000$ or $1.02^x$?

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

Flashcard 35: Which grows faster for large $x$: $x^2+1000$ or $1.02^x$?

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

Flashcard 36: What is the degree of the polynomial $p(x)=7x^5-3x^2+1$ used in comparisons?

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

Flashcard 37: If a table shows $f(0)=5$, $f(1)=9$, $f(2)=13$, what is the growth type?

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

Flashcard 38: Which condition on $b$ gives exponential growth for $f(x)=a\cdot b^x$ with $a>0$?

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

Flashcard 39: Which is exponential: $f(x)=5x^3$ or $g(x)=5\cdot 3^x$?

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

Flashcard 40: Which has constant first differences: $f(x)=3x^2$ or $g(x)=2^x$?

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

Flashcard 41: What is the defining feature of exponential growth visible in a table of values?

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

Flashcard 42: What is the common difference for $f(x)= -3x+10$ in a table with step size $1$?

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

Flashcard 43: Which grows faster for large $x$: $2x^2$ or $1.1^x$?

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

Flashcard 44: Which term determines the end behavior of a polynomial $p(x)=a_nx^n+\cdots$ for large $x$?

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

Flashcard 45: In a table, which pattern indicates polynomial (nonlinear) rather than exponential growth?

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

Flashcard 46: Identify the growth type of $f(x)=6\cdot(0.8)^x$.

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

Flashcard 47: If a table shows $f(0)=3$, $f(1)=6$, $f(2)=12$, what is the growth type?

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

Flashcard 48: Which grows faster for large $x$: $x^5$ or $1.0001^x$?

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

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

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

Flashcard 50: Choose which will be larger for sufficiently large $x$: $f(x)=1000x$ or $g(x)=1.1^x$.

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

Flashcard 51: What does it mean for a function to show exponential growth in the form $f(x)=a\cdot b^x$?

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

Flashcard 52: Choose which will be larger for sufficiently large $x$: $f(x)=1000x$ or $g(x)=1.1^x$.

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

Flashcard 53: Which eventually becomes larger as $x\to\infty$: $x^3$ or $2^x$?

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

Flashcard 54: Which has constant first differences: $f(x)=3x^2$ or $g(x)=2^x$?

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

Flashcard 55: Choose which will be larger for sufficiently large $x$: $f(x)=x^2$ or $g(x)=2^x$.

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

Flashcard 56: Which grows faster for large $x$: $5x+7$ or $1.01^x$?

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

Flashcard 57: Which grows faster for large $x$: $x^5$ or $1.0001^x$?

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

Flashcard 58: What is the defining feature of exponential growth visible in a table of values?

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

Flashcard 59: What is the $y$-intercept of $f(x)=a\cdot b^x$?

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

Flashcard 60: Which grows faster for large $x$: $2x^2$ or $1.1^x$?

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

Flashcard 61: Which grows faster for large $x$: $5x+7$ or $1.01^x$?

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

Flashcard 62: If $f(0)=2$, $f(1)=4$, $f(2)=8$, what is an explicit rule $f(x)=a\cdot b^x$?

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

Flashcard 63: Which condition on $b$ gives exponential decay for $f(x)=a\cdot b^x$ with $a>0$?

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

Flashcard 64: What is the defining feature of linear growth visible in a table of values?

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

Flashcard 65: What is the key CCSS.F-LE.3 comparison statement between exponential and polynomial growth?

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

Flashcard 66: What is one reliable table-based method to decide if data are exponential rather than linear?

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

Flashcard 67: Which eventually becomes larger as $x\to\infty$: $x^3$ or $2^x$?

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

Flashcard 68: What is the common ratio for $f(x)=12\cdot 0.9^x$ in a table with step size $1$?

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

Flashcard 69: What does it mean for a function to show linear growth in the form $f(x)=mx+b$?

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

Flashcard 70: What is the key shape difference on a graph between exponential and linear growth?

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

Flashcard 71: If a table shows $f(0)=3$, $f(1)=6$, $f(2)=12$, what is the growth type?

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

Flashcard 72: For $f(x)=mx+b$, what is $f(x+1)-f(x)$?

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

Flashcard 73: Identify the growth type of $f(x)=6\cdot(1.2)^x$.

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

Flashcard 74: What does it mean for a function to show exponential growth in the form $f(x)=a\cdot b^x$?

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

Flashcard 75: What is the growth type of $f(x)=x^4-2x+9$?

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

Flashcard 76: Which condition on $b$ gives exponential growth for $f(x)=a\cdot b^x$ with $a>0$?

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

Flashcard 77: For $f(x)=a\cdot b^x$ with $a>0$, what is $\frac{f(x+1)}{f(x)}$?

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

Flashcard 78: What is the key CCSS.F-LE.3 comparison statement between exponential and polynomial growth?

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

Flashcard 79: What does it mean for a function to show linear growth in the form $f(x)=mx+b$?

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

Flashcard 80: What is the standard form of a polynomial function used for growth comparisons?

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

Flashcard 81: If $f(0)=7$ and the common ratio is $3$, what is $f(2)$ for $f(x)=a\cdot b^x$?

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

Flashcard 82: What is the key shape difference on a graph between exponential and quadratic growth?

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