Graph Exponential, Logarithmic, and Trig Functions - Algebra 2
Card 1 of 30
What is the amplitude of $y=2\sin(x)-5$?
What is the amplitude of $y=2\sin(x)-5$?
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$2$. Coefficient magnitude is $|2|=2$.
$2$. Coefficient magnitude is $|2|=2$.
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State the five key points for one cycle of $y=\cos(x)$ on $[0,2\pi]$.
State the five key points for one cycle of $y=\cos(x)$ on $[0,2\pi]$.
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$(0,1),(\frac{\pi}{2},0),(\pi,-1),(\frac{3\pi}{2},0),(2\pi,1)$. Cosine starts at maximum, crosses axis, reaches minimum, returns.
$(0,1),(\frac{\pi}{2},0),(\pi,-1),(\frac{3\pi}{2},0),(2\pi,1)$. Cosine starts at maximum, crosses axis, reaches minimum, returns.
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Identify the end behavior of $f(x)= -2\cdot 4^x+1$ as $x\to\infty$.
Identify the end behavior of $f(x)= -2\cdot 4^x+1$ as $x\to\infty$.
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As $x\to\infty$, $f(x)\to-\infty$. Negative coefficient with growing base: approaches negative infinity.
As $x\to\infty$, $f(x)\to-\infty$. Negative coefficient with growing base: approaches negative infinity.
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Find the $x$-intercept of $f(x)=2^x-8$.
Find the $x$-intercept of $f(x)=2^x-8$.
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$(3,0)$. Set $f(x)=0$: $2^x=8=2^3$ gives $x=3$.
$(3,0)$. Set $f(x)=0$: $2^x=8=2^3$ gives $x=3$.
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What is the midline of $y=-3\sin(2x)+4$?
What is the midline of $y=-3\sin(2x)+4$?
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$y=4$. Vertical shift parameter is $k=4$.
$y=4$. Vertical shift parameter is $k=4$.
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What is the period of $y=\sin(Bx)$ if $B$ is negative?
What is the period of $y=\sin(Bx)$ if $B$ is negative?
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Period: $\frac{2\pi}{|B|}$. Absolute value of $B$ gives same period regardless of sign.
Period: $\frac{2\pi}{|B|}$. Absolute value of $B$ gives same period regardless of sign.
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Find the $x$-intercept of $f(x)=\log_2(x)-3$.
Find the $x$-intercept of $f(x)=\log_2(x)-3$.
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$(8,0)$. Set $f(x)=0$: $\log_2(x)-3=0$ gives $x=2^3=8$.
$(8,0)$. Set $f(x)=0$: $\log_2(x)-3=0$ gives $x=2^3=8$.
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State the five key points for one cycle of $y=\sin(x)$ on $[0,2\pi]$.
State the five key points for one cycle of $y=\sin(x)$ on $[0,2\pi]$.
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$(0,0),(\frac{\pi}{2},1),(\pi,0),(\frac{3\pi}{2},-1),(2\pi,0)$. Sine starts at origin, peaks at $\frac{\pi}{2}$, completes cycle.
$(0,0),(\frac{\pi}{2},1),(\pi,0),(\frac{3\pi}{2},-1),(2\pi,0)$. Sine starts at origin, peaks at $\frac{\pi}{2}$, completes cycle.
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Identify the end behavior of $f(x)= -2\cdot 4^x+1$ as $x\to-\infty$.
Identify the end behavior of $f(x)= -2\cdot 4^x+1$ as $x\to-\infty$.
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As $x\to-\infty$, $f(x)\to 1$. As $x$ decreases, $4^x$ approaches 0, so $f(x)$ approaches 1.
As $x\to-\infty$, $f(x)\to 1$. As $x$ decreases, $4^x$ approaches 0, so $f(x)$ approaches 1.
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Identify the end behavior of $f(x)=3\log_2(x-5)+1$ as $x\to 5^+$.
Identify the end behavior of $f(x)=3\log_2(x-5)+1$ as $x\to 5^+$.
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As $x\to 5^+$, $f(x)\to-\infty$. Positive coefficient: as $x$ approaches 5 from right, logarithm decreases.
As $x\to 5^+$, $f(x)\to-\infty$. Positive coefficient: as $x$ approaches 5 from right, logarithm decreases.
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Identify the end behavior of $f(x)=3\log_2(x-5)+1$ as $x\to\infty$.
Identify the end behavior of $f(x)=3\log_2(x-5)+1$ as $x\to\infty$.
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As $x\to\infty$, $f(x)\to\infty$. Positive coefficient with increasing argument: function grows without bound.
As $x\to\infty$, $f(x)\to\infty$. Positive coefficient with increasing argument: function grows without bound.
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What transformation does $f(x)=\log_b(x)+k$ represent relative to $y=\log_b(x)$?
What transformation does $f(x)=\log_b(x)+k$ represent relative to $y=\log_b(x)$?
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Vertical shift by $k$. Output transformation: adding shifts graph up.
Vertical shift by $k$. Output transformation: adding shifts graph up.
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Identify the end behavior of $f(x)=b^x$ when $b>1$ as $x\to\infty$ and $x\to-\infty$.
Identify the end behavior of $f(x)=b^x$ when $b>1$ as $x\to\infty$ and $x\to-\infty$.
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As $x\to\infty$, $f(x)\to\infty$; as $x\to-\infty$, $f(x)\to 0$. For $b>1$: grows to infinity, approaches zero from left.
As $x\to\infty$, $f(x)\to\infty$; as $x\to-\infty$, $f(x)\to 0$. For $b>1$: grows to infinity, approaches zero from left.
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What is the $x$-intercept of $f(x)=\log_b(x)$?
What is the $x$-intercept of $f(x)=\log_b(x)$?
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$(1,0)$. When $x=1$, $\log_b(1)=0$, giving point $(1,0)$.
$(1,0)$. When $x=1$, $\log_b(1)=0$, giving point $(1,0)$.
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What is the $y$-intercept of $f(x)=b^x$?
What is the $y$-intercept of $f(x)=b^x$?
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$(0,1)$. When $x=0$, $b^0=1$, giving point $(0,1)$.
$(0,1)$. When $x=0$, $b^0=1$, giving point $(0,1)$.
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What is the range of $f(x)=a\log_b(x-h)+k$ if $a\neq 0$?
What is the range of $f(x)=a\log_b(x-h)+k$ if $a\neq 0$?
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Range: $(-\infty,\infty)$. Logarithmic functions have unrestricted output values.
Range: $(-\infty,\infty)$. Logarithmic functions have unrestricted output values.
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What is the range of $f(x)=a,b^{(x-h)}+k$ if $a\neq 0$?
What is the range of $f(x)=a,b^{(x-h)}+k$ if $a\neq 0$?
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Range: $(k,\infty)$ if $a>0$; $( -\infty,k)$ if $a<0$. Sign of $a$ determines if range extends above or below asymptote.
Range: $(k,\infty)$ if $a>0$; $( -\infty,k)$ if $a<0$. Sign of $a$ determines if range extends above or below asymptote.
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What is the domain of $f(x)=a\log_b(x-h)+k$?
What is the domain of $f(x)=a\log_b(x-h)+k$?
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Domain: $(h,\infty)$. Argument must be positive, so $x>h$.
Domain: $(h,\infty)$. Argument must be positive, so $x>h$.
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What is the inverse relationship between $y=b^x$ and $y=\log_b(x)$?
What is the inverse relationship between $y=b^x$ and $y=\log_b(x)$?
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They are inverses; their graphs reflect across $y=x$. Exponential and log functions undo each other.
They are inverses; their graphs reflect across $y=x$. Exponential and log functions undo each other.
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Find the horizontal asymptote of $f(x)=3\cdot 2^{x}-5$.
Find the horizontal asymptote of $f(x)=3\cdot 2^{x}-5$.
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$y=-5$. Vertical shift parameter $k=-5$ gives asymptote.
$y=-5$. Vertical shift parameter $k=-5$ gives asymptote.
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Find the vertical asymptote of $f(x)=2\log_3(x+4)-1$.
Find the vertical asymptote of $f(x)=2\log_3(x+4)-1$.
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$x=-4$. Horizontal shift parameter: $x+4=0$ gives $x=-4$.
$x=-4$. Horizontal shift parameter: $x+4=0$ gives $x=-4$.
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Identify whether $f(x)=\left(\frac{1}{3}\right)^x$ shows growth or decay.
Identify whether $f(x)=\left(\frac{1}{3}\right)^x$ shows growth or decay.
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Decay. Base $\frac{1}{3}<1$ causes exponential decay.
Decay. Base $\frac{1}{3}<1$ causes exponential decay.
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Identify whether $f(x)=5\cdot 1.2^x$ shows growth or decay.
Identify whether $f(x)=5\cdot 1.2^x$ shows growth or decay.
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Growth. Base $1.2>1$ causes exponential growth.
Growth. Base $1.2>1$ causes exponential growth.
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Find the $y$-intercept of $f(x)=4\cdot 3^x-7$.
Find the $y$-intercept of $f(x)=4\cdot 3^x-7$.
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$(0,-3)$. At $x=0$: $f(0)=4(1)-7=-3$.
$(0,-3)$. At $x=0$: $f(0)=4(1)-7=-3$.
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What is the domain of $f(x)=a,b^{(x-h)}+k$?
What is the domain of $f(x)=a,b^{(x-h)}+k$?
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Domain: $(-\infty,\infty)$. Exponential functions are defined for all real numbers.
Domain: $(-\infty,\infty)$. Exponential functions are defined for all real numbers.
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What is the vertical asymptote of $f(x)=a\log_b(x-h)+k$?
What is the vertical asymptote of $f(x)=a\log_b(x-h)+k$?
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$x=h$. Horizontal shift parameter $h$ determines vertical asymptote.
$x=h$. Horizontal shift parameter $h$ determines vertical asymptote.
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What is the horizontal asymptote of $f(x)=a,b^{(x-h)}+k$?
What is the horizontal asymptote of $f(x)=a,b^{(x-h)}+k$?
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$y=k$. Vertical shift parameter $k$ determines horizontal asymptote.
$y=k$. Vertical shift parameter $k$ determines horizontal asymptote.
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What is the general form of a logarithmic function used for graphing transformations?
What is the general form of a logarithmic function used for graphing transformations?
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$f(x)=a\log_b(x-h)+k$ with $b>0$ and $b\neq 1$. Standard form showing all transformations with base restrictions.
$f(x)=a\log_b(x-h)+k$ with $b>0$ and $b\neq 1$. Standard form showing all transformations with base restrictions.
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What is the general form of an exponential function used for graphing transformations?
What is the general form of an exponential function used for graphing transformations?
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$f(x)=a,b^{(x-h)}+k$ with $b>0$ and $b\neq 1$. Standard form showing all transformations with base restrictions.
$f(x)=a,b^{(x-h)}+k$ with $b>0$ and $b\neq 1$. Standard form showing all transformations with base restrictions.
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Find the $y$-intercept of $f(x)=\log_5(x+1)$.
Find the $y$-intercept of $f(x)=\log_5(x+1)$.
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$(0,0)$. At $x=0$: $f(0)=\log_5(1)=0$.
$(0,0)$. At $x=0$: $f(0)=\log_5(1)=0$.
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