Algebra 2 Flashcards: Interpreting Sketching Key Features Of Functions

What this deck covers

This deck focuses on Interpreting Sketching Key Features Of 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 slope interpretation of average rate of change on $[a,b]$?

Answer: It is the slope of the secant line: $\frac{f(b)-f(a)}{b-a}$. Geometric interpretation of average rate of change.

Flashcard 2: What is the definition of the $x$-intercept of a function’s graph?

Answer: A point where $y=0$ on the graph, so $f(x)=0$. Where the graph crosses the $x$-axis.

Flashcard 3: Which key feature is identified by evaluating $f(0)$?

Answer: The $y$-intercept $(0,f(0))$. Found by evaluating the function at zero.

Flashcard 4: Which key feature is identified by solving $f(x)=0$?

Answer: The $x$-intercepts (zeros) of the function. Found by setting the function equal to zero.

Flashcard 5: What does the statement “$f$ is increasing on $(a,b)$” mean using inequalities?

Answer: If $x_1<x_2$ in $(a,b)$, then $f(x_1)<f(x_2)$. Definition using ordered pairs in the interval.

Flashcard 6: What does the statement “$f$ is positive on $(a,b)$” mean?

Answer: For all $x$ in $(a,b)$, $f(x)>0$. All function values are above the $x$-axis.

Flashcard 7: Identify the symmetry of $f(x)=\frac{1}{x}$.

Answer: Odd; it has origin symmetry. Standard reciprocal function has origin symmetry.

Flashcard 8: What is the end behavior of $f(x)=\frac{1}{x-3}$ as $x\to\pm\infty$?

Answer: As $x\to\pm\infty$, $f(x)\to 0$. Rational function approaches horizontal asymptote.

Flashcard 9: Identify the horizontal asymptote of $f(x)=\frac{1}{x-3}$.

Answer: Horizontal asymptote: $y=0$. Rational function approaches zero as $x$ grows.

Flashcard 10: Identify the vertical asymptote of $f(x)=\frac{1}{x-3}$.

Answer: Vertical asymptote: $x=3$. Denominator cannot equal zero.

Flashcard 11: Identify the domain restriction implied by the model $f(x)=\sqrt{x-4}$.

Answer: Domain: $x\ge 4$. Expression under square root must be non-negative.

Flashcard 12: Which value is the $y$-intercept of $f(x)=\sqrt{x+1}$?

Answer: The $y$-intercept is $(0,1)$. Evaluate $f(0)=\sqrt{0+1}=1$.

Flashcard 13: Identify the open intervals where $f(x)=(x-2)(x-5)$ is positive.

Answer: $(-\infty,2)\cup(5,\infty)$. Parabola is positive outside its roots.

Flashcard 14: Identify the open interval where $f(x)=(x-2)(x-5)$ is negative.

Answer: $(2,5)$. Parabola is negative between its roots.

Flashcard 15: Identify the average rate of change from $x=2$ to $x=5$ given $f(2)=1$ and $f(5)=7$.

Answer: Average rate of change is $\frac{7-1}{5-2}=2$. Formula: $\frac{\text{change in output}}{\text{change in input}}$.

Flashcard 16: What does the point $(a,b)$ in a table for $f$ represent in context?

Answer: When input is $a$, the output is $f(a)=b$. Input-output relationship in tabular form.

Flashcard 17: Identify the period of $y=\cos(\frac{x}{3})$.

Answer: The period is $6\pi$. Period formula: $\frac{2\pi}{1/3}=6\pi$.

Flashcard 18: Identify the period of $y=\sin(4x)$.

Answer: The period is $\frac{\pi}{2}$. Period formula: $\frac{2\pi}{4}=\frac{\pi}{2}$.

Flashcard 19: What is the midline of $y=3\sin(x)+2$?

Answer: The midline is $y=2$. Vertical shift of the sine function.

Flashcard 20: What is the amplitude of $y=3\sin(x)$?

Answer: The amplitude is $3$. Coefficient of sine function.

Flashcard 21: Identify whether $f(x)=x^2+x$ is even, odd, or neither.

Answer: Neither. Neither even nor odd function.

Flashcard 22: Which statement gives the end behavior of $f(x)=-3x^3+2$?

Answer: As $x\to\infty$, $f(x)\to-\infty$; as $x\to-\infty$, $f(x)\to\infty$. Odd degree with negative leading coefficient.

Flashcard 23: Which statement gives the end behavior of $f(x)=x^4-2x^2$?

Answer: As $x\to\pm\infty$, $f(x)\to\infty$. Even degree with positive leading coefficient.

Flashcard 24: Identify the vertex of $f(x)=x^2-6x+10$.

Answer: The vertex is $(3,1)$. From vertex form $(x-3)^2+1$.

Flashcard 25: Identify the axis of symmetry of $f(x)=x^2-6x+10$.

Answer: The axis of symmetry is $x=3$. Complete the square: $(x-3)^2+1$.

Flashcard 26: On which interval is $f(x)=-2(x+1)^2+7$ decreasing?

Answer: Decreasing on $(-1,\infty)$. Right side of downward-opening parabola.

Flashcard 27: On which interval is $f(x)=-2(x+1)^2+7$ increasing?

Answer: Increasing on $(-\infty,-1)$. Left side of downward-opening parabola.

Flashcard 28: On which interval is $f(x)=(x-3)^2+5$ increasing?

Answer: Increasing on $(3,\infty)$. Right side of upward-opening parabola.

Flashcard 29: On which interval is $f(x)=(x-3)^2+5$ decreasing?

Answer: Decreasing on $(-\infty,3)$. Left side of upward-opening parabola.

Flashcard 30: Identify the relative maximum point of $f(x)=-2(x+1)^2+7$.

Answer: The relative maximum is at $(-1,7)$. Parabola opens downward with vertex at $(-1,7)$.