Algebra 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.

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: For $f(x)=-x^3$, what is the end behavior as $x\to\infty$ and $x\to-\infty$?

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

Flashcard 2: For $f(x)=(x-1)(x-5)$, on which intervals is $f(x)$ positive?

Answer: $(-\infty,1)\cup(5,\infty)$. Parabola opens upward, so it's positive outside zeros.

Flashcard 3: What is the axis of symmetry of $f(x)=(x-3)^2-1$?

Answer: $x=3$. From $(x-3)^2$, the axis is at $x=3$.

Flashcard 4: For $f(x)=x^3$, what is the end behavior as $x\to\infty$ and $x\to-\infty$?

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

Flashcard 5: What is end behavior describing for a function's graph?

Answer: What happens to $f(x)$ as $x\to\infty$ and $x\to-\infty$. Behavior as $x$ approaches positive and negative infinity.

Flashcard 6: Which symmetry does $f(x)=x^3$ have: $y$-axis, origin, or none?

Answer: Origin symmetry (odd). Since $f(-x)=(-x)^3=-x^3=-f(x)$, it's odd.

Flashcard 7: What is the $y$-intercept of a function in terms of the graph?

Answer: The point where the graph crosses the $y$-axis, at $x=0$. Occurs when the input equals zero.

Flashcard 8: For $f(x)=x^2$, what is the end behavior as $x\to\pm\infty$?

Answer: $f(x)\to\infty$ as $x\to\infty$ and as $x\to-\infty$. Upward-opening parabola goes to infinity both ways.

Flashcard 9: What is the meaning of the statement "$f(x)$ is decreasing on $(a,b)$"?

Answer: For $a<x_1<x_2<b$, $f(x_1)>f(x_2)$. Larger inputs give smaller outputs throughout the interval.

Flashcard 10: What is a relative maximum point on a graph?

Answer: A point higher than nearby points (local highest value). A peak on the graph, higher than surrounding points.

Flashcard 11: For $f(x)=(x-2)^2+5$, what is the relative minimum value?

Answer: $5$. The vertex form shows $k$ is the minimum value.

Flashcard 12: For $f(x)=x^2-4$, on which intervals is the function positive?

Answer: $(-\infty,-2)\cup(2,\infty)$. Function is positive outside its zeros at $x=\pm 2$.

Flashcard 13: For $f(x)=(x-1)(x-5)$, on which interval is $f(x)$ negative?

Answer: $(1,5)$. Parabola opens upward, so it's negative between zeros.

Flashcard 14: Identify the $y$-intercept from the table point $(0,7)$.

Answer: $7$. The $y$-intercept is the output value when input is zero.

Flashcard 15: For $g(x)=f(x)-3$, how does the $y$-intercept change from $f(0)$?

Answer: It becomes $f(0)-3$. Subtracting $3$ from $f(0)$ gives the new intercept.

Flashcard 16: Which interval shows $f(x)=2x+1$ is increasing?

Answer: All real numbers, $(-\infty,\infty)$. Linear functions with positive slope increase everywhere.

Flashcard 17: For $g(x)=f(x+2)$, how does the graph shift relative to $f(x)$?

Answer: Shift left $2$ units. Adding inside parentheses shifts left, not right.

Flashcard 18: What is the $y$-intercept of $f(x)=3x-6$?

Answer: $-6$. Substitute $x=0$ into $f(x)=3(0)-6=-6$.

Flashcard 19: For $f(x)=-x^2$, what is the end behavior as $x\to\pm\infty$?

Answer: $f(x)\to-\infty$ as $x\to\infty$ and as $x\to-\infty$. Downward-opening parabola goes to negative infinity both ways.

Flashcard 20: What does a table tell you about the $y$-intercept?

Answer: It is the output when $x=0$, the ordered pair $(0,f(0))$. Find the row where input is zero.

Flashcard 21: If $f(x)$ has period $6$, what is $f(x+6)$ equal to?

Answer: $f(x)$. Since period is $6$, $f(x+6)=f(x)$ by definition.

Flashcard 22: For $f(x)=x^2-4$, what are the $x$-intercepts?

Answer: $x=-2$ and $x=2$. Set $f(x)=0$: $x^2-4=0$, so $x=\pm 2$.

Flashcard 23: For $f(x)=x^2-4$, what is the $y$-intercept?

Answer: $-4$. Substitute $x=0$: $f(0)=0^2-4=-4$.

Flashcard 24: What is a relative minimum point on a graph?

Answer: A point lower than nearby points (local lowest value). A valley on the graph, lower than surrounding points.

Flashcard 25: Which transformation occurs when $f(x)$ becomes $f(x)+k$?

Answer: Vertical shift up $k$ units (down if $k<0$). Adds constant to all output values.

Flashcard 26: What is the axis of symmetry of $f(x)=(x-h)^2+k$?

Answer: $x=h$. Vertical line through the vertex of a parabola.

Flashcard 27: What does it mean for a function to be decreasing on an interval?

Answer: As $x$ increases, $f(x)$ decreases on that interval. The function values fall as you move right.

Flashcard 28: If $f(x)$ has period $4$, what is $f(x+12)$ equal to?

Answer: $f(x)$. Since $12=3\times 4$, we get $f(x+12)=f(x)$.

Flashcard 29: What does it mean for a function to be negative on an interval?

Answer: $f(x)<0$ on that interval (graph is below the $x$-axis). All function values are less than zero.

Flashcard 30: Which transformation occurs when $f(x)$ becomes $f(x-h)$?

Answer: Horizontal shift right $h$ units (left if $h<0$). Replaces $x$ with $x-h$ in the function.

Flashcard 31: For $f(x)=(x-2)^2+5$, what is the $x$-value of the relative minimum?

Answer: $2$. The vertex form shows $h$ is the $x$-coordinate.

Flashcard 32: What does it mean for a function to be increasing on an interval?

Answer: As $x$ increases, $f(x)$ increases on that interval. The function values rise as you move right.

Flashcard 33: For $f(x)=-(x+1)^2+4$, what is the relative maximum value?

Answer: $4$. Negative coefficient creates downward parabola with maximum $k$.

Flashcard 34: What is the meaning of the statement "$f(x)$ is increasing on $(a,b)$"?

Answer: For $a<x_1<x_2<b$, $f(x_1)<f(x_2)$. Larger inputs give larger outputs throughout the interval.

Flashcard 35: What does it mean for a function to be even in terms of symmetry?

Answer: Symmetric about the $y$-axis; $f(-x)=f(x)$. Folding across $y$-axis gives identical graph.

Flashcard 36: What is the axis of symmetry of $f(x)=-(x+4)^2+9$?

Answer: $x=-4$. From $(x+4)^2$, the axis is at $x=-4$.

Flashcard 37: What does it mean for a function to be odd in terms of symmetry?

Answer: Symmetric about the origin; $f(-x)=-f(x)$. Rotating $180°$ about origin gives identical graph.

Flashcard 38: Identify an $x$-intercept from the table point $(-3,0)$.

Answer: $-3$. The $x$-intercept is the input value when output is zero.

Flashcard 39: Identify the $y$-intercept from the table point $(0,7)$.

Answer: $7$. The $y$-intercept is the output value when input is zero.

Flashcard 40: For $f(x)=-(x+1)^2+4$, what is the $x$-value of the relative maximum?

Answer: $-1$. The vertex occurs at $x=h$ from the form.

Flashcard 41: What is the $x$-intercept of $f(x)=3x-6$?

Answer: $2$. Set $f(x)=0$: $3x-6=0$, so $x=2$.

Flashcard 42: What does it mean for a function to be positive on an interval?

Answer: $f(x)>0$ on that interval (graph is above the $x$-axis). All function values are greater than zero.

Flashcard 43: What key features must be labeled when sketching from a verbal description?

Answer: Intercepts, extrema, increasing/decreasing, sign, end behavior, period. Essential elements for complete function analysis.

Flashcard 44: What is the definition of period $P$ for a periodic function?

Answer: The smallest $P>0$ such that $f(x+P)=f(x)$. Adding the period returns the same output.

Flashcard 45: Which interval shows $f(x)=-2x+1$ is decreasing?

Answer: All real numbers, $(-\infty,\infty)$. Linear functions with negative slope decrease everywhere.

Flashcard 46: Which symmetry does $f(x)=x^2-1$ have: $y$-axis, origin, or none?

Answer: $y$-axis symmetry (even). Since $f(-x)=(-x)^2-1=x^2-1=f(x)$, it's even.

Flashcard 47: What is periodicity for a function?

Answer: A repeating pattern with some positive period $P$. Function values repeat at regular intervals.

Flashcard 48: What does a table tell you about an $x$-intercept?

Answer: It occurs where the output is $0$, at a point $(x,0)$. Find the row where output is zero.

Flashcard 49: What is the $x$-intercept of a function in terms of the graph?

Answer: A point where the graph crosses the $x$-axis, where $y=0$. Occurs when the output equals zero.

Flashcard 50: For $f(x)=(x-1)(x-5)$, what are the $x$-intercepts?

Answer: $x=1$ and $x=5$. Set each factor to zero: $x-1=0$ or $x-5=0$.

Flashcard 51: What is the axis of symmetry of $f(x)=(x-3)^2-1$?

Answer: $x=3$. From $(x-3)^2$, the axis is at $x=3$.

Flashcard 52: Identify an $x$-intercept from the table point $(-3,0)$.

Answer: $-3$. The $x$-intercept is the input value when output is zero.

Flashcard 53: If $f(x)$ has period $4$, what is $f(x+12)$ equal to?

Answer: $f(x)$. Since $12=3\times 4$, we get $f(x+12)=f(x)$.

Flashcard 54: For $f(x)=x^2-4$, on which interval is the function negative?

Answer: $(-2,2)$. Function is negative between its zeros at $x=\pm 2$.