Graph Square Root and Piecewise Functions - Algebra
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What is the range of the parent square root function $f(x)=\sqrt{x}$?
What is the range of the parent square root function $f(x)=\sqrt{x}$?
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$y\ge 0$. Square root output is always non-negative.
$y\ge 0$. Square root output is always non-negative.
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What is the vertex form of an absolute value function used for graphing?
What is the vertex form of an absolute value function used for graphing?
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$f(x)=a|x-h|+k$. Standard form showing vertex $(h,k)$ and vertical stretch/compression $a$.
$f(x)=a|x-h|+k$. Standard form showing vertex $(h,k)$ and vertical stretch/compression $a$.
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What is the transformation form for a square root function used for graphing?
What is the transformation form for a square root function used for graphing?
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$f(x)=a\sqrt{x-h}+k$. Standard form showing starting point $(h,k)$ and vertical stretch/compression $a$.
$f(x)=a\sqrt{x-h}+k$. Standard form showing starting point $(h,k)$ and vertical stretch/compression $a$.
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What is the transformation form for a cube root function used for graphing?
What is the transformation form for a cube root function used for graphing?
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$f(x)=a\sqrt[3]{x-h}+k$. Standard form showing inflection point $(h,k)$ and vertical stretch/compression $a$.
$f(x)=a\sqrt[3]{x-h}+k$. Standard form showing inflection point $(h,k)$ and vertical stretch/compression $a$.
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What point is the starting point of $f(x)=a\sqrt{x-h}+k$ on its graph?
What point is the starting point of $f(x)=a\sqrt{x-h}+k$ on its graph?
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$(h,k)$. Square root function begins at this translated point.
$(h,k)$. Square root function begins at this translated point.
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What point is the center point of $f(x)=a\sqrt[3]{x-h}+k$ on its graph?
What point is the center point of $f(x)=a\sqrt[3]{x-h}+k$ on its graph?
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$(h,k)$. Cube root function passes through this translated point.
$(h,k)$. Cube root function passes through this translated point.
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Identify the horizontal shift of $f(x)=\sqrt{x-5}$ compared to $\sqrt{x}$.
Identify the horizontal shift of $f(x)=\sqrt{x-5}$ compared to $\sqrt{x}$.
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Shift right $5$. Inside the radical, $-5$ shifts right by $5$ units.
Shift right $5$. Inside the radical, $-5$ shifts right by $5$ units.
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Identify the vertical shift of $f(x)=\sqrt{x}+3$ compared to $\sqrt{x}$.
Identify the vertical shift of $f(x)=\sqrt{x}+3$ compared to $\sqrt{x}$.
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Shift up $3$. Adding outside the function shifts up by $3$ units.
Shift up $3$. Adding outside the function shifts up by $3$ units.
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Identify the reflection in $f(x)=-\sqrt{x}$ compared to $\sqrt{x}$.
Identify the reflection in $f(x)=-\sqrt{x}$ compared to $\sqrt{x}$.
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Reflect over the $x$-axis. Negative sign in front flips graph over the $x$-axis.
Reflect over the $x$-axis. Negative sign in front flips graph over the $x$-axis.
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Identify the vertical shift of $f(x)=\sqrt[3]{x}-4$ compared to $\sqrt[3]{x}$.
Identify the vertical shift of $f(x)=\sqrt[3]{x}-4$ compared to $\sqrt[3]{x}$.
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Shift down $4$. Subtracting outside the function shifts down by $4$ units.
Shift down $4$. Subtracting outside the function shifts down by $4$ units.
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Identify the reflection in $f(x)=\sqrt[3]{-x}$ compared to $\sqrt[3]{x}$.
Identify the reflection in $f(x)=\sqrt[3]{-x}$ compared to $\sqrt[3]{x}$.
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Reflect over the $y$-axis. Negative inside the radical flips graph over the $y$-axis.
Reflect over the $y$-axis. Negative inside the radical flips graph over the $y$-axis.
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Identify the reflection in $f(x)=-\sqrt[3]{x}$ compared to $\sqrt[3]{x}$.
Identify the reflection in $f(x)=-\sqrt[3]{x}$ compared to $\sqrt[3]{x}$.
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Reflect over the $x$-axis. Negative sign in front flips graph over the $x$-axis.
Reflect over the $x$-axis. Negative sign in front flips graph over the $x$-axis.
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Identify the horizontal shift of $f(x)=|x+7|$ compared to $|x|$.
Identify the horizontal shift of $f(x)=|x+7|$ compared to $|x|$.
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Shift left $7$. Inside absolute value, $+7$ shifts left by $7$ units.
Shift left $7$. Inside absolute value, $+7$ shifts left by $7$ units.
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Identify the vertical shift of $f(x)=|x|-2$ compared to $|x|$.
Identify the vertical shift of $f(x)=|x|-2$ compared to $|x|$.
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Shift down $2$. Subtracting outside the function shifts down by $2$ units.
Shift down $2$. Subtracting outside the function shifts down by $2$ units.
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What is the domain of $f(x)=\sqrt{x-9}$?
What is the domain of $f(x)=\sqrt{x-9}$?
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$x\ge 9$. Square root requires $x-9\ge 0$.
$x\ge 9$. Square root requires $x-9\ge 0$.
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Identify the reflection in $f(x)=-|x|$ compared to $|x|$.
Identify the reflection in $f(x)=-|x|$ compared to $|x|$.
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Reflect over the $x$-axis. Negative sign in front flips graph over the $x$-axis.
Reflect over the $x$-axis. Negative sign in front flips graph over the $x$-axis.
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What is the domain of $f(x)=\sqrt{2x+6}$?
What is the domain of $f(x)=\sqrt{2x+6}$?
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$x\ge -3$. Square root requires $2x+6\ge 0$, so $x\ge -3$.
$x\ge -3$. Square root requires $2x+6\ge 0$, so $x\ge -3$.
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What is the range of $f(x)=\sqrt{x}+5$?
What is the range of $f(x)=\sqrt{x}+5$?
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$y\ge 5$. Adding $5$ shifts all output values up by $5$.
$y\ge 5$. Adding $5$ shifts all output values up by $5$.
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What is the range of $f(x)=-\sqrt{x}+2$?
What is the range of $f(x)=-\sqrt{x}+2$?
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$y\le 2$. Negative flips and adding $2$ gives maximum value $2$.
$y\le 2$. Negative flips and adding $2$ gives maximum value $2$.
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What is the domain of $f(x)=|x-4|+1$?
What is the domain of $f(x)=|x-4|+1$?
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$(-\infty,\infty)$. Absolute value functions accept all real inputs.
$(-\infty,\infty)$. Absolute value functions accept all real inputs.
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What point is the vertex of $f(x)=a|x-h|+k$ on its graph?
What point is the vertex of $f(x)=a|x-h|+k$ on its graph?
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$(h,k)$. Absolute value function has its corner at this translated point.
$(h,k)$. Absolute value function has its corner at this translated point.
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What is the range of $f(x)=-|x+2|-3$?
What is the range of $f(x)=-|x+2|-3$?
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$y\le -3$. Negative with vertex at $(-2,-3)$ gives maximum value $-3$.
$y\le -3$. Negative with vertex at $(-2,-3)$ gives maximum value $-3$.
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What is the vertex of $f(x)=|x-6|+4$?
What is the vertex of $f(x)=|x-6|+4$?
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$(6,4)$. Vertex form parameters: $(h,k) = (6,4)$.
$(6,4)$. Vertex form parameters: $(h,k) = (6,4)$.
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What is the starting point of $f(x)=\sqrt{x+1}-3$?
What is the starting point of $f(x)=\sqrt{x+1}-3$?
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$(-1,-3)$. Transformation parameters: $(h,k) = (-1,-3)$.
$(-1,-3)$. Transformation parameters: $(h,k) = (-1,-3)$.
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What is the center point of $f(x)=\sqrt[3]{x-8}+2$?
What is the center point of $f(x)=\sqrt[3]{x-8}+2$?
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$(8,2)$. Transformation parameters: $(h,k) = (8,2)$.
$(8,2)$. Transformation parameters: $(h,k) = (8,2)$.
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What are two symmetric line pieces that define $|x|$ as a piecewise function?
What are two symmetric line pieces that define $|x|$ as a piecewise function?
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$|x|=\begin{cases}x&x\ge 0\\-x&x<0\end{cases}$. Absolute value splits into two linear pieces at $x=0$.
$|x|=\begin{cases}x&x\ge 0\\-x&x<0\end{cases}$. Absolute value splits into two linear pieces at $x=0$.
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What is the definition of a piecewise function in terms of rules and intervals?
What is the definition of a piecewise function in terms of rules and intervals?
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A function defined by different rules on different intervals. Each piece has a specific domain interval.
A function defined by different rules on different intervals. Each piece has a specific domain interval.
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What is the domain of the parent cube root function $f(x)=\sqrt[3]{x}$?
What is the domain of the parent cube root function $f(x)=\sqrt[3]{x}$?
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$(-\infty,\infty)$. Cube root accepts all real number inputs.
$(-\infty,\infty)$. Cube root accepts all real number inputs.
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What is the vertex of the parent absolute value function $f(x)=|x|$?
What is the vertex of the parent absolute value function $f(x)=|x|$?
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$(0,0)$. V-shape opens at the origin.
$(0,0)$. V-shape opens at the origin.
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What do you use to decide which rule to apply in a piecewise function?
What do you use to decide which rule to apply in a piecewise function?
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The input $x$ and the stated interval conditions. Check which interval contains the input value.
The input $x$ and the stated interval conditions. Check which interval contains the input value.
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