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: Which interval is $f(x)=x^2-4$ positive on?

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

Flashcard 2: What does it mean for a function to be increasing on an interval?

Answer: As $x$ increases, $f(x)$ increases on that interval. The graph slopes upward from left to right.

Flashcard 3: 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 4: On which interval is $f(x)=-2(x+1)^2+7$ decreasing?

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

Flashcard 5: What is the vertex of a parabola written as $y=a(x-h)^2+k$?

Answer: The vertex is $(h,k)$. The turning point of the parabola.

Flashcard 6: Identify the open interval where $f(x)=(x-2)(x-5)$ is negative.

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

Flashcard 7: 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 8: What does it mean for a function to be odd?

Answer: It satisfies $f(-x)=-f(x)$ and has origin symmetry. Point reflection through the origin.

Flashcard 9: On which interval is $f(x)=(x-3)^2+5$ increasing?

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

Flashcard 10: What is the meaning of end behavior of a function?

Answer: How $f(x)$ behaves as $x\to\infty$ and as $x\to-\infty$. Describes the function's behavior at extreme $x$-values.

Flashcard 11: Identify the maximum value of $f(x)=-2(x+1)^2+7$.

Answer: Maximum value is $7$. Vertex gives the maximum for downward parabola.

Flashcard 12: Identify the relative minimum point of $f(x)=(x-3)^2+5$.

Answer: The relative minimum is at $(3,5)$. Parabola opens upward with vertex at $(3,5)$.

Flashcard 13: What does it mean for a function to be decreasing on an interval?

Answer: As $x$ increases, $f(x)$ decreases on that interval. The graph slopes downward from left to right.

Flashcard 14: 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 15: What is a relative maximum point on a graph?

Answer: A point where $f(x)$ is greatest compared to nearby $x$-values. A local peak on the graph.

Flashcard 16: Identify the $y$-intercept of $f(x)=2x-6$.

Answer: The $y$-intercept is $(0,-6)$. Evaluate $f(0)=2(0)-6=-6$.

Flashcard 17: Identify the $x$-intercept of $f(x)=2x-6$.

Answer: The $x$-intercept is $(3,0)$. Set $2x-6=0$ and solve for $x$.

Flashcard 18: Identify the $x$-intercepts of $f(x)=(x-1)(x+4)$.

Answer: The $x$-intercepts are $(1,0)$ and $(-4,0)$. Set each factor equal to zero.

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

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

Flashcard 21: What is the period of $y=\sin(bx)$ for $b>0$?

Answer: The period is $\frac{2\pi}{b}$. Coefficient $b$ compresses the period by factor $b$.

Flashcard 22: What is the definition of periodicity for a function?

Answer: There exists $T>0$ such that $f(x+T)=f(x)$ for all $x$. The function repeats its pattern every $T$ units.

Flashcard 23: 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 24: Which interval is $f(x)=x^2-4$ negative on?

Answer: $(-2,2)$. Parabola is negative between the roots.

Flashcard 25: What is the definition of the $y$-intercept of a function's graph?

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

Flashcard 26: Identify the $y$-intercept of $f(x)=(x-1)(x+4)$.

Answer: The $y$-intercept is $(0,-4)$. Evaluate $f(0)=(0-1)(0+4)=-4$.

Flashcard 27: What is the period of $y=\cos(x)$?

Answer: The period is $2\pi$. Standard period for cosine function.

Flashcard 28: On which interval is $f(x)=-2(x+1)^2+7$ increasing?

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

Flashcard 29: Identify the period of $y=\sin(4x)$.

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

Flashcard 30: What is a relative minimum point on a graph?

Answer: A point where $f(x)$ is least compared to nearby $x$-values. A local valley on the graph.

Flashcard 31: Identify the vertical asymptote of $f(x)=\frac{1}{x-3}$.

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

Flashcard 32: Identify the symmetry of $f(x)=\frac{1}{x}$.

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

Flashcard 33: 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 34: What is the meaning of $f(a)>0$ in terms of the graph at $x=a$?

Answer: The graph is above the $x$-axis at $x=a$. Positive function values mean points above the axis.

Flashcard 35: 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 36: What is the amplitude of $y=3\sin(x)$?

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

Flashcard 37: What is the axis of symmetry for a parabola written as $y=a(x-h)^2+k$?

Answer: The axis of symmetry is $x=h$. Vertical line through the vertex of the parabola.

Flashcard 38: Identify the vertex of $f(x)=x^2-6x+10$.

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

Flashcard 39: 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 40: 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 41: 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 42: 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 43: 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 44: What is the meaning of $f(a)<0$ in terms of the graph at $x=a$?

Answer: The graph is below the $x$-axis at $x=a$. Negative function values mean points below the axis.

Flashcard 45: 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 46: On which interval is $f(x)=(x-3)^2+5$ decreasing?

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

Flashcard 47: What does it mean for a function to be even?

Answer: It satisfies $f(-x)=f(x)$ and is symmetric about the $y$-axis. Mirror image across the $y$-axis.

Flashcard 48: What is the period of $y=\sin(x)$?

Answer: The period is $2\pi$. Standard period for sine function.

Flashcard 49: Which key feature is identified by evaluating $f(0)$?

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

Flashcard 50: 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 51: What is the period of $y=\cos(bx)$ for $b>0$?

Answer: The period is $\frac{2\pi}{b}$. Coefficient $b$ compresses the period by factor $b$.

Flashcard 52: 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 53: 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 $(-1,7)$.

Flashcard 54: Identify whether $f(x)=x^2+x$ is even, odd, or neither.

Answer: Neither. Neither even nor odd function.

Flashcard 55: Identify the axis of symmetry of $f(x)=x^2-6x+10$.

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