Comparing Linear, Quadratic, Polynomial, Exponential Growth - Algebra 2
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What is the key table test for linear growth using outputs $y$ at equal $x$-steps?
What is the key table test for linear growth using outputs $y$ at equal $x$-steps?
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Successive differences $y_{k+1}-y_k$ are constant. Linear growth shows constant additive differences in tables.
Successive differences $y_{k+1}-y_k$ are constant. Linear growth shows constant additive differences in tables.
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What is the simplest inequality statement for “exponential eventually beats polynomial” for $b>1$?
What is the simplest inequality statement for “exponential eventually beats polynomial” for $b>1$?
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There exists $N$ such that $x>N\Rightarrow b^x>p(x)$. General statement that exponential eventually dominates any polynomial.
There exists $N$ such that $x>N\Rightarrow b^x>p(x)$. General statement that exponential eventually dominates any polynomial.
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What condition on $b$ makes $f(x)=ab^x$ an increasing exponential function?
What condition on $b$ makes $f(x)=ab^x$ an increasing exponential function?
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$b>1$ (with $a>0$). Base must exceed 1 for exponential growth (increasing function).
$b>1$ (with $a>0$). Base must exceed 1 for exponential growth (increasing function).
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What condition on $r$ makes $f(x)=a(1+r)^x$ represent exponential growth?
What condition on $r$ makes $f(x)=a(1+r)^x$ represent exponential growth?
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$r>0$ (and $a>0$). Growth rate must be positive for $(1+r)^x$ to represent increasing exponential.
$r>0$ (and $a>0$). Growth rate must be positive for $(1+r)^x$ to represent increasing exponential.
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What is the constant multiplicative factor between consecutive outputs of $f(x)=ab^x$?
What is the constant multiplicative factor between consecutive outputs of $f(x)=ab^x$?
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$ rac{f(x+1)}{f(x)}=b$. Consecutive outputs have constant multiplicative ratio equal to base $b$.
$ rac{f(x+1)}{f(x)}=b$. Consecutive outputs have constant multiplicative ratio equal to base $b$.
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What is the constant additive difference between consecutive outputs of a linear function $f(x)=mx+b$?
What is the constant additive difference between consecutive outputs of a linear function $f(x)=mx+b$?
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$f(x+1)-f(x)=m$. Linear functions have constant additive differences equal to slope $m$.
$f(x+1)-f(x)=m$. Linear functions have constant additive differences equal to slope $m$.
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Which grows faster as $x\to\infty$: $f(x)=b^x$ with $b>1$ or $g(x)=x^n$?
Which grows faster as $x\to\infty$: $f(x)=b^x$ with $b>1$ or $g(x)=x^n$?
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$b^x$ grows faster than $x^n$. Exponential functions with base $b>1$ eventually dominate any polynomial.
$b^x$ grows faster than $x^n$. Exponential functions with base $b>1$ eventually dominate any polynomial.
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What limit statement expresses that exponential growth eventually exceeds polynomial growth?
What limit statement expresses that exponential growth eventually exceeds polynomial growth?
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$\lim_{x\to\infty}\frac{x^n}{b^x}=0$ for $b>1$. Ratio of polynomial to exponential approaches zero as exponential dominates.
$\lim_{x\to\infty}\frac{x^n}{b^x}=0$ for $b>1$. Ratio of polynomial to exponential approaches zero as exponential dominates.
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What does “eventually exceeds” mean for functions $f$ and $g$ as $x$ increases?
What does “eventually exceeds” mean for functions $f$ and $g$ as $x$ increases?
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There exists $N$ such that $x>N\Rightarrow f(x)>g(x)$. Formal definition: after some threshold $N$, $f$ is always greater than $g$.
There exists $N$ such that $x>N\Rightarrow f(x)>g(x)$. Formal definition: after some threshold $N$, $f$ is always greater than $g$.
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What is the general form of a polynomial function used in comparisons with exponentials?
What is the general form of a polynomial function used in comparisons with exponentials?
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$p(x)=a_nx^n+\cdots+a_1x+a_0$ with $a_n\neq 0$. Standard polynomial form with leading coefficient $a_n$ nonzero.
$p(x)=a_nx^n+\cdots+a_1x+a_0$ with $a_n\neq 0$. Standard polynomial form with leading coefficient $a_n$ nonzero.
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What is the degree of the polynomial $p(x)=7x^5-2x^3+9$?
What is the degree of the polynomial $p(x)=7x^5-2x^3+9$?
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$5$. Degree is the highest power of $x$ in the polynomial.
$5$. Degree is the highest power of $x$ in the polynomial.
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Which has constant ratio: exponential $ab^x$ or polynomial $x^n$?
Which has constant ratio: exponential $ab^x$ or polynomial $x^n$?
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Exponential $ab^x$. Exponential functions have constant ratios between consecutive outputs.
Exponential $ab^x$. Exponential functions have constant ratios between consecutive outputs.
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Which has constant first differences: linear $mx+b$ or exponential $ab^x$?
Which has constant first differences: linear $mx+b$ or exponential $ab^x$?
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Linear $mx+b$. Linear functions have constant first differences between consecutive outputs.
Linear $mx+b$. Linear functions have constant first differences between consecutive outputs.
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For $f(x)=3\cdot 2^x$, what is the growth factor per $1$ unit increase in $x$?
For $f(x)=3\cdot 2^x$, what is the growth factor per $1$ unit increase in $x$?
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$2$. Growth factor is the base $b$ in exponential function $ab^x$.
$2$. Growth factor is the base $b$ in exponential function $ab^x$.
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For $f(x)=5(1.08)^x$, what is the percent increase per $1$ unit of $x$?
For $f(x)=5(1.08)^x$, what is the percent increase per $1$ unit of $x$?
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$8%$. Growth rate is $(1.08-1)\times 100% = 8%$.
$8%$. Growth rate is $(1.08-1)\times 100% = 8%$.
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Identify whether $f(x)=4x+7$ is linear, polynomial (nonlinear), or exponential.
Identify whether $f(x)=4x+7$ is linear, polynomial (nonlinear), or exponential.
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Linear. Form $mx+b$ indicates linear function (degree 1).
Linear. Form $mx+b$ indicates linear function (degree 1).
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Identify whether $f(x)=2x^2-3x+1$ is linear, polynomial (nonlinear), or exponential.
Identify whether $f(x)=2x^2-3x+1$ is linear, polynomial (nonlinear), or exponential.
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Polynomial (quadratic). Quadratic polynomial has degree 2 (highest power is $x^2$).
Polynomial (quadratic). Quadratic polynomial has degree 2 (highest power is $x^2$).
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Identify whether $f(x)=7\cdot(1.5)^x$ is linear, polynomial, or exponential.
Identify whether $f(x)=7\cdot(1.5)^x$ is linear, polynomial, or exponential.
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Exponential. Form $ab^x$ with constant base indicates exponential function.
Exponential. Form $ab^x$ with constant base indicates exponential function.
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Which grows faster for large $x$: $f(x)=2^x$ or $g(x)=100x^3$?
Which grows faster for large $x$: $f(x)=2^x$ or $g(x)=100x^3$?
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$2^x$. Exponential with base $>1$ eventually exceeds any polynomial.
$2^x$. Exponential with base $>1$ eventually exceeds any polynomial.
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Which grows faster for large $x$: $f(x)=1.01^x$ or $g(x)=x^{10}$?
Which grows faster for large $x$: $f(x)=1.01^x$ or $g(x)=x^{10}$?
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$1.01^x$. Even small exponential bases $>1$ eventually dominate high-degree polynomials.
$1.01^x$. Even small exponential bases $>1$ eventually dominate high-degree polynomials.
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Which grows faster for large $x$: $f(x)=3^x$ or $g(x)=x^2+10x$?
Which grows faster for large $x$: $f(x)=3^x$ or $g(x)=x^2+10x$?
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$3^x$. Exponential with base $>1$ eventually exceeds any polynomial.
$3^x$. Exponential with base $>1$ eventually exceeds any polynomial.
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Which grows faster for large $x$: $f(x)=x^5$ or $g(x)=10x^2$?
Which grows faster for large $x$: $f(x)=x^5$ or $g(x)=10x^2$?
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$x^5$. Higher degree polynomial grows faster than lower degree polynomial.
$x^5$. Higher degree polynomial grows faster than lower degree polynomial.
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What is the key table test for exponential growth using outputs $y$ at equal $x$-steps?
What is the key table test for exponential growth using outputs $y$ at equal $x$-steps?
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Successive ratios $\frac{y_{k+1}}{y_k}$ are constant. Exponential growth shows constant multiplicative ratios in tables.
Successive ratios $\frac{y_{k+1}}{y_k}$ are constant. Exponential growth shows constant multiplicative ratios in tables.
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What is the key table test for linear growth using outputs $y$ at equal $x$-steps?
What is the key table test for linear growth using outputs $y$ at equal $x$-steps?
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Successive differences $y_{k+1}-y_k$ are constant. Linear growth shows constant additive differences in tables.
Successive differences $y_{k+1}-y_k$ are constant. Linear growth shows constant additive differences in tables.
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In a table, if $y$ values are $3,6,12,24$ for consecutive $x$, what type of growth is shown?
In a table, if $y$ values are $3,6,12,24$ for consecutive $x$, what type of growth is shown?
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Exponential (ratio $2$). Each value doubles the previous (constant ratio of 2).
Exponential (ratio $2$). Each value doubles the previous (constant ratio of 2).
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In a table, if $y$ values are $5,9,13,17$ for consecutive $x$, what type of growth is shown?
In a table, if $y$ values are $5,9,13,17$ for consecutive $x$, what type of growth is shown?
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Linear (difference $4$). Each value increases by 4 from previous (constant difference).
Linear (difference $4$). Each value increases by 4 from previous (constant difference).
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In a table, if $y$ values are $1,4,9,16$ for $x=1,2,3,4$, what type of function fits exactly?
In a table, if $y$ values are $1,4,9,16$ for $x=1,2,3,4$, what type of function fits exactly?
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Quadratic ($y=x^2$). Values are perfect squares: $1^2, 2^2, 3^2, 4^2$.
Quadratic ($y=x^2$). Values are perfect squares: $1^2, 2^2, 3^2, 4^2$.
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What does it suggest if first differences are not constant but second differences are constant?
What does it suggest if first differences are not constant but second differences are constant?
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Quadratic growth. Constant second differences indicate quadratic (degree 2) polynomial.
Quadratic growth. Constant second differences indicate quadratic (degree 2) polynomial.
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For $f(x)=ab^x$, what is $f(0)$ in terms of $a$?
For $f(x)=ab^x$, what is $f(0)$ in terms of $a$?
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$f(0)=a$. Any number to power 0 equals 1, so $b^0=1$ and $f(0)=a\cdot 1=a$.
$f(0)=a$. Any number to power 0 equals 1, so $b^0=1$ and $f(0)=a\cdot 1=a$.
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For $f(x)=7\cdot 3^x$, what is $f(0)$?
For $f(x)=7\cdot 3^x$, what is $f(0)$?
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$7$. Substitute $x=0$: $f(0)=7\cdot 3^0=7\cdot 1=7$.
$7$. Substitute $x=0$: $f(0)=7\cdot 3^0=7\cdot 1=7$.
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