Algebra Flashcards: Graph Square Root And Piecewise Functions

What this deck covers

This deck focuses on Graph Square Root And Piecewise 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: What is the range of $f(x)=|x-4|+1$?

Answer: $y\ge 1$. Vertex at $(4,1)$ gives minimum output value $1$.

Flashcard 2: What is the vertex of $f(x)=|x-6|+4$?

Answer: $(6,4)$. Vertex form parameters: $(h,k) = (6,4)$.

Flashcard 3: What is the parent function for a cube root graph?

Answer: $f(x)=\sqrt[3]{x}$. Basic cube root function before any transformations.

Flashcard 4: Evaluate $\lfloor 2.9\rfloor$.

Answer: $2$. Greatest integer $\le 2.9$ is $2$.

Flashcard 5: What is the $y$-intercept of $f(x)=\sqrt{x}$?

Answer: $(0,0)$. Evaluate at $x=0$: $f(0)=\sqrt{0}=0$.

Flashcard 6: Identify the horizontal shift of $f(x)=|x+7|$ compared to $|x|$.

Answer: Shift left $7$. Inside absolute value, $+7$ shifts left by $7$ units.

Flashcard 7: Identify the reflection in $f(x)=-\sqrt{x}$ compared to $\sqrt{x}$.

Answer: Reflect over the $x$-axis. Negative sign in front flips graph over the $x$-axis.

Flashcard 8: Identify whether $x> -1$ should have an open or closed circle at $x=-1$.

Answer: Open circle at $x=-1$. The $>$ symbol excludes the endpoint.

Flashcard 9: What is the transformation form for a cube root function used for graphing?

Answer: $f(x)=a\sqrt[3]{x-h}+k$. Standard form showing inflection point $(h,k)$ and vertical stretch/compression $a$.

Flashcard 10: Identify the vertical shift of $f(x)=\sqrt{x}+3$ compared to $\sqrt{x}$.

Answer: Shift up $3$. Adding outside the function shifts up by $3$ units.

Flashcard 11: What does a closed circle at an endpoint mean on a piecewise graph?

Answer: Endpoint is included (uses $\le$ or $\ge$). Inequality symbols determine endpoint inclusion.

Flashcard 12: Identify the horizontal shift of $f(x)=|x+7|$ compared to $|x|$.

Answer: Shift left $7$. Inside absolute value, $+7$ shifts left by $7$ units.

Flashcard 13: Identify the horizontal shift of $f(x)=\sqrt{x-5}$ compared to $\sqrt{x}$.

Answer: Shift right $5$. Inside the radical, $-5$ shifts right by $5$ units.

Flashcard 14: Identify the reflection in $f(x)=\sqrt[3]{-x}$ compared to $\sqrt[3]{x}$.

Answer: Reflect over the $y$-axis. Negative inside the radical flips graph over the $y$-axis.

Flashcard 15: Identify the reflection in $f(x)=-|x|$ compared to $|x|$.

Answer: Reflect over the $x$-axis. Negative sign in front flips graph over the $x$-axis.

Flashcard 16: What point is the vertex of $f(x)=a|x-h|+k$ on its graph?

Answer: $(h,k)$. Absolute value function has its corner at this translated point.

Flashcard 17: Identify the reflection in $f(x)=-\sqrt{x}$ compared to $\sqrt{x}$.

Answer: Reflect over the $x$-axis. Negative sign in front flips graph over the $x$-axis.

Flashcard 18: What is the range of the parent cube root function $f(x)=\sqrt[3]{x}$?

Answer: $(-\infty,\infty)$. Cube root produces all real number outputs.

Flashcard 19: Evaluate $\lfloor -3\rfloor$.

Answer: $-3$. Greatest integer $\le -3$ is $-3$ itself.

Flashcard 20: What is the parent function for a square root graph?

Answer: $f(x)=\sqrt{x}$. Basic square root function before any transformations.

Flashcard 21: Identify whether $x\le 2$ should have an open or closed circle at $x=2$.

Answer: Closed circle at $x=2$. The $\le$ symbol includes the endpoint.

Flashcard 22: What does an open circle at an endpoint mean on a piecewise graph?

Answer: Endpoint is not included (uses $<$ or $>$). Inequality symbols determine endpoint inclusion.

Flashcard 23: What is the parent function for a square root graph?

Answer: $f(x)=\sqrt{x}$. Basic square root function before any transformations.

Flashcard 24: What is the domain of $f(x)=|x-4|+1$?

Answer: $(-\infty,\infty)$. Absolute value functions accept all real inputs.

Flashcard 25: What is the vertex form of an absolute value function used for graphing?

Answer: $f(x)=a|x-h|+k$. Standard form showing vertex $(h,k)$ and vertical stretch/compression $a$.

Flashcard 26: What point is the vertex of $f(x)=a|x-h|+k$ on its graph?

Answer: $(h,k)$. Absolute value function has its corner at this translated point.

Flashcard 27: Identify the horizontal shift of $f(x)=\sqrt[3]{x+2}$ compared to $\sqrt[3]{x}$.

Answer: Shift left $2$. Inside the radical, $+2$ shifts left by $2$ units.

Flashcard 28: Identify the horizontal shift of $f(x)=\sqrt[3]{x+2}$ compared to $\sqrt[3]{x}$.

Answer: Shift left $2$. Inside the radical, $+2$ shifts left by $2$ units.

Flashcard 29: What is the vertex of the parent absolute value function $f(x)=|x|$?

Answer: $(0,0)$. V-shape opens at the origin.

Flashcard 30: What do you use to decide which rule to apply in a piecewise function?

Answer: The input $x$ and the stated interval conditions. Check which interval contains the input value.

Flashcard 31: What is the range of $f(x)=\sqrt{x}+5$?

Answer: $y\ge 5$. Adding $5$ shifts all output values up by $5$.

Flashcard 32: What is the vertex form of an absolute value function used for graphing?

Answer: $f(x)=a|x-h|+k$. Standard form showing vertex $(h,k)$ and vertical stretch/compression $a$.

Flashcard 33: What is the vertex of the parent absolute value function $f(x)=|x|$?

Answer: $(0,0)$. V-shape opens at the origin.

Flashcard 34: What is the starting point of $f(x)=\sqrt{x+1}-3$?

Answer: $(-1,-3)$. Transformation parameters: $(h,k) = (-1,-3)$.

Flashcard 35: Identify the reflection in $f(x)=-|x|$ compared to $|x|$.

Answer: Reflect over the $x$-axis. Negative sign in front flips graph over the $x$-axis.

Flashcard 36: What is the step function notation for the greatest integer function?

Answer: $f(x)=\lfloor x\rfloor$. Floor function symbol for step functions.

Flashcard 37: What is the range of $f(x)=-|x+2|-3$?

Answer: $y\le -3$. Negative with vertex at $(-2,-3)$ gives maximum value $-3$.

Flashcard 38: Identify the reflection in $f(x)=-\sqrt[3]{x}$ compared to $\sqrt[3]{x}$.

Answer: Reflect over the $x$-axis. Negative sign in front flips graph over the $x$-axis.

Flashcard 39: What is the range of $f(x)=-\sqrt{x}+2$?

Answer: $y\le 2$. Negative flips and adding $2$ gives maximum value $2$.

Flashcard 40: What is the definition of a piecewise function in terms of rules and intervals?

Answer: A function defined by different rules on different intervals. Each piece has a specific domain interval.

Flashcard 41: What is the parent function for an absolute value graph?

Answer: $f(x)=|x|$. Basic absolute value function before any transformations.

Flashcard 42: What is the range of $f(x)=-\sqrt{x}+2$?

Answer: $y\le 2$. Negative flips and adding $2$ gives maximum value $2$.

Flashcard 43: What is the output of the ceiling function $\lceil x\rceil$ in words?

Answer: The least integer greater than or equal to $x$. Always rounds up to the nearest integer.

Flashcard 44: What is the domain of the parent square root function $f(x)=\sqrt{x}$?

Answer: $x\ge 0$. Square root requires non-negative input values.

Flashcard 45: What point is the center point of $f(x)=a\sqrt[3]{x-h}+k$ on its graph?

Answer: $(h,k)$. Cube root function passes through this translated point.

Flashcard 46: Identify the horizontal shift of $f(x)=\sqrt{x-5}$ compared to $\sqrt{x}$.

Answer: Shift right $5$. Inside the radical, $-5$ shifts right by $5$ units.

Flashcard 47: What is the transformation form for a square root function used for graphing?

Answer: $f(x)=a\sqrt{x-h}+k$. Standard form showing starting point $(h,k)$ and vertical stretch/compression $a$.

Flashcard 48: What is the range of $f(x)=|x-4|+1$?

Answer: $y\ge 1$. Vertex at $(4,1)$ gives minimum output value $1$.

Flashcard 49: Evaluate $\lceil -1.2\rceil$.

Answer: $-1$. Least integer $\ge -1.2$ is $-1$.

Flashcard 50: What is the range of the parent square root function $f(x)=\sqrt{x}$?

Answer: $y\ge 0$. Square root output is always non-negative.

Flashcard 51: What point is the starting point of $f(x)=a\sqrt{x-h}+k$ on its graph?

Answer: $(h,k)$. Square root function begins at this translated point.

Flashcard 52: What is the output of the ceiling function $\lceil x\rceil$ in words?

Answer: The least integer greater than or equal to $x$. Always rounds up to the nearest integer.

Flashcard 53: What is the domain of $f(x)=|x-4|+1$?

Answer: $(-\infty,\infty)$. Absolute value functions accept all real inputs.

Flashcard 54: What do you use to decide which rule to apply in a piecewise function?

Answer: The input $x$ and the stated interval conditions. Check which interval contains the input value.

Flashcard 55: What is the center point of $f(x)=\sqrt[3]{x-8}+2$?

Answer: $(8,2)$. Transformation parameters: $(h,k) = (8,2)$.

Flashcard 56: Evaluate $\lceil 2.1\rceil$.

Answer: $3$. Least integer $\ge 2.1$ is $3$.

Flashcard 57: What is the domain of $f(x)=\sqrt{2x+6}$?

Answer: $x\ge -3$. Square root requires $2x+6\ge 0$, so $x\ge -3$.

Flashcard 58: What is the range of the parent square root function $f(x)=\sqrt{x}$?

Answer: $y\ge 0$. Square root output is always non-negative.

Flashcard 59: Evaluate $\lfloor -1.2\rfloor$.

Answer: $-2$. Greatest integer $\le -1.2$ is $-2$.

Flashcard 60: Identify the vertical shift of $f(x)=|x|-2$ compared to $|x|$.

Answer: Shift down $2$. Subtracting outside the function shifts down by $2$ units.

Flashcard 61: What is the domain of $f(x)=\sqrt{2x+6}$?

Answer: $x\ge -3$. Square root requires $2x+6\ge 0$, so $x\ge -3$.

Flashcard 62: Evaluate $\lfloor -1.2\rfloor$.

Answer: $-2$. Greatest integer $\le -1.2$ is $-2$.

Flashcard 63: What is the range of the parent cube root function $f(x)=\sqrt[3]{x}$?

Answer: $(-\infty,\infty)$. Cube root produces all real number outputs.

Flashcard 64: Evaluate $\lceil 2.1\rceil$.

Answer: $3$. Least integer $\ge 2.1$ is $3$.

Flashcard 65: What is the starting point of $f(x)=\sqrt{x+1}-3$?

Answer: $(-1,-3)$. Transformation parameters: $(h,k) = (-1,-3)$.

Flashcard 66: What point is the starting point of $f(x)=a\sqrt{x-h}+k$ on its graph?

Answer: $(h,k)$. Square root function begins at this translated point.

Flashcard 67: What is the domain of the parent cube root function $f(x)=\sqrt[3]{x}$?

Answer: $(-\infty,\infty)$. Cube root accepts all real number inputs.

Flashcard 68: What is the center point of $f(x)=\sqrt[3]{x-8}+2$?

Answer: $(8,2)$. Transformation parameters: $(h,k) = (8,2)$.

Flashcard 69: What is the $x$-intercept of $f(x)=\sqrt{x-4}$?

Answer: $(4,0)$. Set $f(x)=0$ and solve: $\sqrt{x-4}=0$ gives $x=4$.

Flashcard 70: Identify whether $x> -1$ should have an open or closed circle at $x=-1$.

Answer: Open circle at $x=-1$. The $>$ symbol excludes the endpoint.

Flashcard 71: What is the domain of the parent square root function $f(x)=\sqrt{x}$?

Answer: $x\ge 0$. Square root requires non-negative input values.

Flashcard 72: What is the domain of the parent cube root function $f(x)=\sqrt[3]{x}$?

Answer: $(-\infty,\infty)$. Cube root accepts all real number inputs.

Flashcard 73: Identify the vertical shift of $f(x)=\sqrt[3]{x}-4$ compared to $\sqrt[3]{x}$.

Answer: Shift down $4$. Subtracting outside the function shifts down by $4$ units.

Flashcard 74: Identify the reflection in $f(x)=-\sqrt[3]{x}$ compared to $\sqrt[3]{x}$.

Answer: Reflect over the $x$-axis. Negative sign in front flips graph over the $x$-axis.

Flashcard 75: What is the range of $f(x)=-|x+2|-3$?

Answer: $y\le -3$. Negative with vertex at $(-2,-3)$ gives maximum value $-3$.

Flashcard 76: What is the domain of $f(x)=\sqrt{x-9}$?

Answer: $x\ge 9$. Square root requires $x-9\ge 0$.

Flashcard 77: Identify whether $x\le 2$ should have an open or closed circle at $x=2$.

Answer: Closed circle at $x=2$. The $\le$ symbol includes the endpoint.

Flashcard 78: Identify the reflection in $f(x)=\sqrt[3]{-x}$ compared to $\sqrt[3]{x}$.

Answer: Reflect over the $y$-axis. Negative inside the radical flips graph over the $y$-axis.

Flashcard 79: Identify the vertical shift of $f(x)=|x|-2$ compared to $|x|$.

Answer: Shift down $2$. Subtracting outside the function shifts down by $2$ units.

Flashcard 80: What is the transformation form for a cube root function used for graphing?

Answer: $f(x)=a\sqrt[3]{x-h}+k$. Standard form showing inflection point $(h,k)$ and vertical stretch/compression $a$.

Flashcard 81: What is the vertex of $f(x)=|x-6|+4$?

Answer: $(6,4)$. Vertex form parameters: $(h,k) = (6,4)$.

Flashcard 82: What is the parent function for a cube root graph?

Answer: $f(x)=\sqrt[3]{x}$. Basic cube root function before any transformations.

Flashcard 83: Evaluate $\lceil -1.2\rceil$.

Answer: $-1$. Least integer $\ge -1.2$ is $-1$.

Flashcard 84: What is the $x$-intercept of $f(x)=\sqrt{x-4}$?

Answer: $(4,0)$. Set $f(x)=0$ and solve: $\sqrt{x-4}=0$ gives $x=4$.

Flashcard 85: What is the output of the greatest integer function $\lfloor x\rfloor$ in words?

Answer: The greatest integer less than or equal to $x$. Always rounds down to the nearest integer.

Flashcard 86: What is the transformation form for a square root function used for graphing?

Answer: $f(x)=a\sqrt{x-h}+k$. Standard form showing starting point $(h,k)$ and vertical stretch/compression $a$.

Flashcard 87: What is the range of $f(x)=\sqrt{x}+5$?

Answer: $y\ge 5$. Adding $5$ shifts all output values up by $5$.

Flashcard 88: Evaluate $\lfloor 2.9\rfloor$.

Answer: $2$. Greatest integer $\le 2.9$ is $2$.

Flashcard 89: What is the step function notation for the greatest integer function?

Answer: $f(x)=\lfloor x\rfloor$. Floor function symbol for step functions.

Flashcard 90: What is the parent function for an absolute value graph?

Answer: $f(x)=|x|$. Basic absolute value function before any transformations.

Flashcard 91: Evaluate $\lfloor -3\rfloor$.

Answer: $-3$. Greatest integer $\le -3$ is $-3$ itself.

Flashcard 92: What does a closed circle at an endpoint mean on a piecewise graph?

Answer: Endpoint is included (uses $\le$ or $\ge$). Inequality symbols determine endpoint inclusion.

Flashcard 93: What does an open circle at an endpoint mean on a piecewise graph?

Answer: Endpoint is not included (uses $<$ or $>$). Inequality symbols determine endpoint inclusion.

Flashcard 94: What is the output of the greatest integer function $\lfloor x\rfloor$ in words?

Answer: The greatest integer less than or equal to $x$. Always rounds down to the nearest integer.

Flashcard 95: What point is the center point of $f(x)=a\sqrt[3]{x-h}+k$ on its graph?

Answer: $(h,k)$. Cube root function passes through this translated point.

Flashcard 96: Identify the vertical shift of $f(x)=\sqrt{x}+3$ compared to $\sqrt{x}$.

Answer: Shift up $3$. Adding outside the function shifts up by $3$ units.

Flashcard 97: What is the definition of a piecewise function in terms of rules and intervals?

Answer: A function defined by different rules on different intervals. Each piece has a specific domain interval.

Flashcard 98: What are two symmetric line pieces that define $|x|$ as a piecewise function?

Answer: $|x|=\begin{cases}x&x\ge 0\\-x&x<0\end{cases}$. Absolute value splits into two linear pieces at $x=0$.

Flashcard 99: Identify the vertical shift of $f(x)=\sqrt[3]{x}-4$ compared to $\sqrt[3]{x}$.

Answer: Shift down $4$. Subtracting outside the function shifts down by $4$ units.

Flashcard 100: What are two symmetric line pieces that define $|x|$ as a piecewise function?

Answer: $|x|=\begin{cases}x&x\ge 0\\-x&x<0\end{cases}$. Absolute value splits into two linear pieces at $x=0$.