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

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

Flashcard 2: 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 3: 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 4: 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 5: What is the midline of $y=-3\sin(2x)+4$?

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

Flashcard 6: 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 7: 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 8: 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 9: 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 10: 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.

Flashcard 11: 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 12: 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 13: 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 14: 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 15: What is the $y$-intercept of $f(x)=b^x$?

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

Flashcard 16: 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 17: 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 18: 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 19: 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 20: Find the horizontal asymptote of $f(x)=3\cdot 2^{x}-5$.

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

Flashcard 21: 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 22: 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 23: Identify whether $f(x)=5\cdot 1.2^x$ shows growth or decay.

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

Flashcard 24: 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 25: 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 26: 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 27: What is the horizontal asymptote of $f(x)=a\,b^{(x-h)}+k$?

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

Flashcard 28: 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 29: 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 30: Find the $y$-intercept of $f(x)=\log_5(x+1)$.

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