Algebra 2 Flashcards: Graph Exponential Logarithmic And Trig Functions

What this deck covers

This deck focuses on Graph Exponential Logarithmic And Trig Functions, 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 transformation does $f(x)=b^{(x-h)}$ represent relative to $y=b^x$?

Answer: Horizontal shift right by $h$ (left if $h<0$). Input transformation: subtracting shifts graph right.

Flashcard 2: State the five key $x$-values for one cycle of $y=\sin(x)$ from $0$ to $2\pi$.

Answer: $0,\,\frac{\pi}{2},\,\pi,\,\frac{3\pi}{2},\,2\pi$. Standard quarter-period intervals for sine function.

Flashcard 3: What is the midline of $y=-3\sin(2x)+4$?

Answer: $y=4$. Vertical shift parameter is $k=4$.

Flashcard 4: What is the amplitude of $y=\frac{1}{2}\cos(x)-1$?

Answer: $\frac{1}{2}$. Coefficient magnitude is $|\frac{1}{2}|=\frac{1}{2}$.

Flashcard 5: What is the general form of an exponential function used for graphing transformations?

Answer: $f(x)=a\,b^{(x-h)}+k$ with $b>0$ and $b\neq 1$. Standard form showing all transformations with base restrictions.

Flashcard 6: What is the period of $y=\sin(3x)$?

Answer: $\frac{2\pi}{3}$. Period formula: $\frac{2\pi}{3}$ when $B=3$.

Flashcard 7: Identify the end behavior of $f(x)= -2\cdot 4^x+1$ as $x\to-\infty$.

Answer: As $x\to-\infty$, $f(x)\to 1$. As $x$ decreases, $4^x$ approaches 0, so $f(x)$ approaches 1.

Flashcard 8: Identify the end behavior of $f(x)=\log_b(x)$ when $b>1$ as $x\to\infty$ and $x\to 0^+$.

Answer: As $x\to\infty$, $f(x)\to\infty$; as $x\to 0^+$, $f(x)\to-\infty$. For $b>1$: grows without bound, decreases without bound near asymptote.

Flashcard 9: What transformation does $f(x)=\log_b(x-h)$ represent relative to $y=\log_b(x)$?

Answer: Horizontal shift right by $h$ (left if $h<0$). Input transformation: subtracting shifts graph right.

Flashcard 10: State the general form for graphing a sinusoid with period, midline, and amplitude labeled.

Answer: $y=A\sin(B(x-h))+k$ or $y=A\cos(B(x-h))+k$. Shows amplitude $A$, period factor $B$, shifts $h,k$.

Flashcard 11: What are the maximum and minimum values of $y=2\sin(x)-5$?

Answer: Max $=-3$, Min $=-7$. Midline $-5$ plus/minus amplitude $2$.

Flashcard 12: Identify the period of $y=7\sin\left(\frac{\pi}{2}x\right)+3$.

Answer: $4$. Period formula: $\frac{2\pi}{\pi/2}=4$ when $B=\frac{\pi}{2}$.

Flashcard 13: What is the general form of a logarithmic function used for graphing transformations?

Answer: $f(x)=a\log_b(x-h)+k$ with $b>0$ and $b\neq 1$. Standard form showing all transformations with base restrictions.

Flashcard 14: What is the period of $y=A\sin(Bx)$ or $y=A\cos(Bx)$?

Answer: Period: $\frac{2\pi}{|B|}$. Period formula: $\frac{2\pi}{|B|}$ for sine and cosine.

Flashcard 15: What is the inverse relationship between $y=b^x$ and $y=\log_b(x)$?

Answer: They are inverses; their graphs reflect across $y=x$. Exponential and log functions undo each other.

Flashcard 16: State the five key points for one cycle of $y=\cos(x)$ on $[0,2\pi]$.

Answer: $(0,1),(\frac{\pi}{2},0),(\pi,-1),(\frac{3\pi}{2},0),(2\pi,1)$. Cosine starts at maximum, crosses axis, reaches minimum, returns.

Flashcard 17: What is the domain of $f(x)=a\log_b(x-h)+k$?

Answer: Domain: $(h,\infty)$. Argument must be positive, so $x>h$.

Flashcard 18: What is the range of $f(x)=a\log_b(x-h)+k$ if $a\neq 0$?

Answer: Range: $(-\infty,\infty)$. Logarithmic functions have unrestricted output values.

Flashcard 19: Identify the end behavior of $f(x)=b^x$ when $b>1$ as $x\to\infty$ and $x\to-\infty$.

Answer: 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.

Flashcard 20: Identify the end behavior of $f(x)= -2\cdot 4^x+1$ as $x\to\infty$.

Answer: As $x\to\infty$, $f(x)\to-\infty$. Negative coefficient with growing base: approaches negative infinity.

Flashcard 21: What is the midline of $y=2\sin(x)-5$?

Answer: $y=-5$. Vertical shift parameter is $k=-5$.

Flashcard 22: What is the $x$-intercept of $f(x)=\log_b(x)$?

Answer: $(1,0)$. When $x=1$, $\log_b(1)=0$, giving point $(1,0)$.

Flashcard 23: Find the $x$-intercept of $f(x)=\log_2(x)-3$.

Answer: $(8,0)$. Set $f(x)=0$: $\log_2(x)-3=0$ gives $x=2^3=8$.

Flashcard 24: What is the amplitude of $y=A\sin(B(x-h))+k$ or $y=A\cos(B(x-h))+k$?

Answer: Amplitude: $|A|$. Coefficient $A$ determines maximum distance from midline.

Flashcard 25: What is the midline of $y=A\sin(B(x-h))+k$ or $y=A\cos(B(x-h))+k$?

Answer: Midline: $y=k$. Vertical shift $k$ determines center line of oscillation.

Flashcard 26: What is the domain of $f(x)=a\,b^{(x-h)}+k$?

Answer: Domain: $(-\infty,\infty)$. Exponential functions are defined for all real numbers.

Flashcard 27: What is the period of $y=\sin(Bx)$ if $B$ is negative?

Answer: Period: $\frac{2\pi}{|B|}$. Absolute value of $B$ gives same period regardless of sign.

Flashcard 28: What is the amplitude of $y=2\sin(x)-5$?

Answer: $2$. Coefficient magnitude is $|2|=2$.

Flashcard 29: What is the period of $y=A\tan(Bx)$?

Answer: Period: $\frac{\pi}{|B|}$. Tangent has period $\pi$, so formula is $\frac{\pi}{|B|}$.

Flashcard 30: What transformation does $f(x)=a\,b^x$ represent relative to $y=b^x$?

Answer: Vertical stretch by $|a|$; reflection across $x$-axis if $a<0$. Coefficient affects vertical scaling and reflection.

Flashcard 31: What is the $y$-intercept of $f(x)=b^x$?

Answer: $(0,1)$. When $x=0$, $b^0=1$, giving point $(0,1)$.

Flashcard 32: Find the horizontal asymptote of $f(x)=3\cdot 2^{x}-5$.

Answer: $y=-5$. Vertical shift parameter $k=-5$ gives asymptote.

Flashcard 33: Identify whether $f(x)=5\cdot 1.2^x$ shows growth or decay.

Answer: Growth. Base $1.2>1$ causes exponential growth.

Flashcard 34: Identify the end behavior of $f(x)=\log_b(x)$ when $0<b<1$ as $x\to\infty$ and $x\to 0^+$.

Answer: As $x\to\infty$, $f(x)\to-\infty$; as $x\to 0^+$, $f(x)\to\infty$. For $0<b<1$: decreases without bound as $x$ increases.

Flashcard 35: What transformation does $f(x)=b^x+k$ represent relative to $y=b^x$?

Answer: Vertical shift by $k$. Output transformation: adding shifts graph up.

Flashcard 36: Find the $y$-intercept of $f(x)=4\cdot 3^x-7$.

Answer: $(0,-3)$. At $x=0$: $f(0)=4(1)-7=-3$.

Flashcard 37: What is the vertical asymptote of $f(x)=a\log_b(x-h)+k$?

Answer: $x=h$. Horizontal shift parameter $h$ determines vertical asymptote.

Flashcard 38: Identify the midline of $y=7\sin\left(\frac{\pi}{2}x\right)+3$.

Answer: $y=3$. Vertical shift parameter is $k=3$.

Flashcard 39: What is the horizontal asymptote of $f(x)=a\,b^{(x-h)}+k$?

Answer: $y=k$. Vertical shift parameter $k$ determines horizontal asymptote.

Flashcard 40: What is the period of $y=\tan(5x)$?

Answer: $\frac{\pi}{5}$. Tangent period formula: $\frac{\pi}{5}$ when $B=5$.

Flashcard 41: What is the midline of $y=\frac{1}{2}\cos(x)-1$?

Answer: $y=-1$. Vertical shift parameter is $k=-1$.

Flashcard 42: What is the range of $y=A\sin(B(x-h))+k$?

Answer: Range: $[k-|A|,\,k+|A|]$. Amplitude $|A|$ above and below midline $k$.

Flashcard 43: Find the $x$-intercept of $f(x)=2^x-8$.

Answer: $(3,0)$. Set $f(x)=0$: $2^x=8=2^3$ gives $x=3$.

Flashcard 44: Identify the amplitude of $y=7\sin\left(\frac{\pi}{2}x\right)+3$.

Answer: $7$. Coefficient magnitude is $|7|=7$.

Flashcard 45: What is the amplitude of $y=-3\sin(2x)+4$?

Answer: $3$. Coefficient magnitude is $|-3|=3$.

Flashcard 46: What transformation does $f(x)=\log_b(x)+k$ represent relative to $y=\log_b(x)$?

Answer: Vertical shift by $k$. Output transformation: adding shifts graph up.

Flashcard 47: Identify the end behavior of $f(x)=3\log_2(x-5)+1$ as $x\to\infty$.

Answer: As $x\to\infty$, $f(x)\to\infty$. Positive coefficient with increasing argument: function grows without bound.

Flashcard 48: Identify the end behavior of $f(x)=b^x$ when $0<b<1$ as $x\to\infty$ and $x\to-\infty$.

Answer: As $x\to\infty$, $f(x)\to 0$; as $x\to-\infty$, $f(x)\to\infty$. For $0<b<1$: decays to zero, grows from left.

Flashcard 49: What is the period of $y=2\cos\left(\frac{1}{4}x\right)$?

Answer: $8\pi$. Period formula: $\frac{2\pi}{1/4}=8\pi$ when $B=\frac{1}{4}$.

Flashcard 50: Find the $y$-intercept of $f(x)=\log_5(x+1)$.

Answer: $(0,0)$. At $x=0$: $f(0)=\log_5(1)=0$.

Flashcard 51: Find the vertical asymptote of $f(x)=2\log_3(x+4)-1$.

Answer: $x=-4$. Horizontal shift parameter: $x+4=0$ gives $x=-4$.

Flashcard 52: What is the range of $y=A\cos(B(x-h))+k$?

Answer: Range: $[k-|A|,\,k+|A|]$. Amplitude $|A|$ above and below midline $k$.

Flashcard 53: Identify whether $f(x)=\left(\frac{1}{3}\right)^x$ shows growth or decay.

Answer: Decay. Base $\frac{1}{3}<1$ causes exponential decay.

Flashcard 54: State the five key points for one cycle of $y=\sin(x)$ on $[0,2\pi]$.

Answer: $(0,0),(\frac{\pi}{2},1),(\pi,0),(\frac{3\pi}{2},-1),(2\pi,0)$. Sine starts at origin, peaks at $\frac{\pi}{2}$, completes cycle.

Flashcard 55: What is the range of $f(x)=a\,b^{(x-h)}+k$ if $a\neq 0$?

Answer: Range: $(k,\infty)$ if $a>0$; $( -\infty,k)$ if $a<0$. Sign of $a$ determines if range extends above or below asymptote.

Flashcard 56: Identify the end behavior of $f(x)=3\log_2(x-5)+1$ as $x\to 5^+$.

Answer: As $x\to 5^+$, $f(x)\to-\infty$. Positive coefficient: as $x$ approaches 5 from right, logarithm decreases.