Comparing Linear, Quadratic, Polynomial, Exponential Growth - Algebra
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Which eventually becomes larger as $x\to\infty$: $x^3$ or $2^x$?
Which eventually becomes larger as $x\to\infty$: $x^3$ or $2^x$?
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$2^x$ eventually becomes larger than $x^3$. Exponential functions eventually dominate polynomial functions.
$2^x$ eventually becomes larger than $x^3$. Exponential functions eventually dominate polynomial functions.
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What is the key shape difference on a graph between exponential and quadratic growth?
What is the key shape difference on a graph between exponential and quadratic growth?
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Exponential eventually becomes steeper than any parabola. Exponential growth rate increases while quadratic growth rate is bounded.
Exponential eventually becomes steeper than any parabola. Exponential growth rate increases while quadratic growth rate is bounded.
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Which has constant first differences: $f(x)=3x^2$ or $g(x)=2^x$?
Which has constant first differences: $f(x)=3x^2$ or $g(x)=2^x$?
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Neither; constant first differences occur for linear functions. Only linear functions have constant first differences in tables.
Neither; constant first differences occur for linear functions. Only linear functions have constant first differences in tables.
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Which has constant ratio in a table: $f(x)=7x+1$ or $g(x)=3\cdot 2^x$?
Which has constant ratio in a table: $f(x)=7x+1$ or $g(x)=3\cdot 2^x$?
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$g(x)=3\cdot 2^x$ has constant ratio $2$. Exponential functions have constant ratios, not linear functions.
$g(x)=3\cdot 2^x$ has constant ratio $2$. Exponential functions have constant ratios, not linear functions.
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What is the key shape difference on a graph between exponential and linear growth?
What is the key shape difference on a graph between exponential and linear growth?
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Exponential curves upward more and more; linear is a straight line. Exponential curves get steeper while linear graphs maintain constant slope.
Exponential curves upward more and more; linear is a straight line. Exponential curves get steeper while linear graphs maintain constant slope.
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Identify the growth type of $f(x)=6\cdot(1.2)^x$.
Identify the growth type of $f(x)=6\cdot(1.2)^x$.
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Exponential growth. Base $1.2 > 1$ indicates exponential growth pattern.
Exponential growth. Base $1.2 > 1$ indicates exponential growth pattern.
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Identify the growth type of $f(x)=6\cdot(0.8)^x$.
Identify the growth type of $f(x)=6\cdot(0.8)^x$.
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Exponential decay. Base $0.8 < 1$ indicates exponential decay pattern.
Exponential decay. Base $0.8 < 1$ indicates exponential decay pattern.
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What does it mean for a function to show exponential growth in the form $f(x)=a\cdot b^x$?
What does it mean for a function to show exponential growth in the form $f(x)=a\cdot b^x$?
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$b>1$ (with $a>0$) so outputs multiply by the same factor each step. Exponential growth multiplies by a constant factor $b$ at each step.
$b>1$ (with $a>0$) so outputs multiply by the same factor each step. Exponential growth multiplies by a constant factor $b$ at each step.
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What does it mean for a function to show linear growth in the form $f(x)=mx+b$?
What does it mean for a function to show linear growth in the form $f(x)=mx+b$?
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It increases by a constant add-on amount for equal $x$-steps. Linear functions have constant slope $m$, adding the same amount each step.
It increases by a constant add-on amount for equal $x$-steps. Linear functions have constant slope $m$, adding the same amount each step.
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What is the key CCSS.F-LE.3 comparison statement between exponential and polynomial growth?
What is the key CCSS.F-LE.3 comparison statement between exponential and polynomial growth?
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Exponential growth eventually exceeds any polynomial growth. This is the core principle of CCSS.F-LE.3 about exponential dominance.
Exponential growth eventually exceeds any polynomial growth. This is the core principle of CCSS.F-LE.3 about exponential dominance.
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Which grows faster for large $x$: $2x^2$ or $1.1^x$?
Which grows faster for large $x$: $2x^2$ or $1.1^x$?
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$1.1^x$ eventually grows faster than $2x^2$. Exponential functions eventually outgrow all polynomial functions.
$1.1^x$ eventually grows faster than $2x^2$. Exponential functions eventually outgrow all polynomial functions.
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Which grows faster for large $x$: $5x+7$ or $1.01^x$?
Which grows faster for large $x$: $5x+7$ or $1.01^x$?
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$1.01^x$ eventually grows faster than $5x+7$. Even small exponential bases like $1.01$ eventually dominate linear functions.
$1.01^x$ eventually grows faster than $5x+7$. Even small exponential bases like $1.01$ eventually dominate linear functions.
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Which grows faster for large $x$: $x^5$ or $1.0001^x$?
Which grows faster for large $x$: $x^5$ or $1.0001^x$?
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$1.0001^x$ eventually grows faster than $x^5$. Exponential growth dominates polynomial growth regardless of degree.
$1.0001^x$ eventually grows faster than $x^5$. Exponential growth dominates polynomial growth regardless of degree.
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What is the defining feature of exponential growth visible in a table of values?
What is the defining feature of exponential growth visible in a table of values?
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A constant ratio $\frac{f(x+1)}{f(x)}=b$ (approximately). Exponential functions have constant ratios between consecutive terms.
A constant ratio $\frac{f(x+1)}{f(x)}=b$ (approximately). Exponential functions have constant ratios between consecutive terms.
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What is the defining feature of linear growth visible in a table of values?
What is the defining feature of linear growth visible in a table of values?
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A constant difference $f(x+1)-f(x)=m$. Linear functions add the same amount $m$ between consecutive terms.
A constant difference $f(x+1)-f(x)=m$. Linear functions add the same amount $m$ between consecutive terms.
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In a table, which pattern indicates polynomial (nonlinear) rather than exponential growth?
In a table, which pattern indicates polynomial (nonlinear) rather than exponential growth?
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Differences change, and ratios are not constant. Polynomial functions have varying differences and ratios between terms.
Differences change, and ratios are not constant. Polynomial functions have varying differences and ratios between terms.
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What is the standard form of a polynomial function used for growth comparisons?
What is the standard form of a polynomial function used for growth comparisons?
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$p(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0$ with $n\ge 1$. Standard polynomial form with degree $n$ and leading coefficient $a_n$.
$p(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0$ with $n\ge 1$. Standard polynomial form with degree $n$ and leading coefficient $a_n$.
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Which term determines the end behavior of a polynomial $p(x)=a_nx^n+\cdots$ for large $x$?
Which term determines the end behavior of a polynomial $p(x)=a_nx^n+\cdots$ for large $x$?
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The leading term $a_nx^n$. The highest degree term dominates behavior for large $x$ values.
The leading term $a_nx^n$. The highest degree term dominates behavior for large $x$ values.
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If $f(2)=12$ and $f(3)=18$ for an exponential model, what is $\frac{f(3)}{f(2)}$?
If $f(2)=12$ and $f(3)=18$ for an exponential model, what is $\frac{f(3)}{f(2)}$?
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$1.5$. For exponential functions, consecutive ratios equal the base.
$1.5$. For exponential functions, consecutive ratios equal the base.
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If $f(0)=7$ and the common ratio is $3$, what is $f(2)$ for $f(x)=a\cdot b^x$?
If $f(0)=7$ and the common ratio is $3$, what is $f(2)$ for $f(x)=a\cdot b^x$?
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$63$. With $a=7$ and $b=3$, we get $f(2)=7 \cdot 3^2=63$.
$63$. With $a=7$ and $b=3$, we get $f(2)=7 \cdot 3^2=63$.
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What is the common difference for $f(x)= -3x+10$ in a table with step size $1$?
What is the common difference for $f(x)= -3x+10$ in a table with step size $1$?
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$-3$. Slope $-3$ means each step subtracts $3$.
$-3$. Slope $-3$ means each step subtracts $3$.
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What is the common ratio for $f(x)=12\cdot 0.9^x$ in a table with step size $1$?
What is the common ratio for $f(x)=12\cdot 0.9^x$ in a table with step size $1$?
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$0.9$. Base $0.9$ means each step multiplies by $0.9$.
$0.9$. Base $0.9$ means each step multiplies by $0.9$.
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Which increases by multiplication: $f(x)=8\cdot(1.3)^x$ or $g(x)=8x+1.3$?
Which increases by multiplication: $f(x)=8\cdot(1.3)^x$ or $g(x)=8x+1.3$?
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$f(x)=8\cdot(1.3)^x$. Exponential form multiplies by base; linear form adds slope.
$f(x)=8\cdot(1.3)^x$. Exponential form multiplies by base; linear form adds slope.
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What is the degree of the polynomial $p(x)=7x^5-3x^2+1$ used in comparisons?
What is the degree of the polynomial $p(x)=7x^5-3x^2+1$ used in comparisons?
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$5$. The highest power of $x$ determines polynomial degree.
$5$. The highest power of $x$ determines polynomial degree.
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What is the growth type of $f(x)=x^4-2x+9$?
What is the growth type of $f(x)=x^4-2x+9$?
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Polynomial growth (degree $4$). Highest power term determines polynomial degree and growth rate.
Polynomial growth (degree $4$). Highest power term determines polynomial degree and growth rate.
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If $f(x)=3\cdot(1.5)^x$ and $g(x)=100x^3$, what is true for large $x$?
If $f(x)=3\cdot(1.5)^x$ and $g(x)=100x^3$, what is true for large $x$?
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$f(x)>g(x)$ for sufficiently large $x$. Exponential functions eventually dominate polynomial functions.
$f(x)>g(x)$ for sufficiently large $x$. Exponential functions eventually dominate polynomial functions.
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If $f(x)=2^x$ and $g(x)=x^2$, what is true about $f(x)-g(x)$ for large $x$?
If $f(x)=2^x$ and $g(x)=x^2$, what is true about $f(x)-g(x)$ for large $x$?
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It becomes positive and keeps increasing. Exponential $f(x)$ eventually exceeds polynomial $g(x)$, so difference grows.
It becomes positive and keeps increasing. Exponential $f(x)$ eventually exceeds polynomial $g(x)$, so difference grows.
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What does it mean if a graph becomes steeper and steeper as $x$ increases?
What does it mean if a graph becomes steeper and steeper as $x$ increases?
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It suggests exponential (or faster-than-linear) growth. Increasing steepness indicates accelerating growth like exponential functions.
It suggests exponential (or faster-than-linear) growth. Increasing steepness indicates accelerating growth like exponential functions.
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What is one reliable table-based method to decide if data are exponential rather than linear?
What is one reliable table-based method to decide if data are exponential rather than linear?
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Check for a constant ratio, not a constant difference. Constant ratios distinguish exponential from linear growth patterns.
Check for a constant ratio, not a constant difference. Constant ratios distinguish exponential from linear growth patterns.
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Choose which will be larger for sufficiently large $x$: $f(x)=x^7$ or $g(x)=1.01^x$.
Choose which will be larger for sufficiently large $x$: $f(x)=x^7$ or $g(x)=1.01^x$.
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$g(x)=1.01^x$. Even small exponential base dominates high-degree polynomial.
$g(x)=1.01^x$. Even small exponential base dominates high-degree polynomial.
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