Interpreting/Sketching Key Features of Functions - Algebra
Card 1 of 30
If $f(x)$ has period $4$, what is $f(x+12)$ equal to?
If $f(x)$ has period $4$, what is $f(x+12)$ equal to?
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$f(x)$. Since $12=3\times 4$, we get $f(x+12)=f(x)$.
$f(x)$. Since $12=3\times 4$, we get $f(x+12)=f(x)$.
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For $f(x)=(x-1)(x-5)$, on which intervals is $f(x)$ positive?
For $f(x)=(x-1)(x-5)$, on which intervals is $f(x)$ positive?
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$(-\infty,1)\cup(5,\infty)$. Parabola opens upward, so it's positive outside zeros.
$(-\infty,1)\cup(5,\infty)$. Parabola opens upward, so it's positive outside zeros.
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For $g(x)=f(x+2)$, how does the graph shift relative to $f(x)$?
For $g(x)=f(x+2)$, how does the graph shift relative to $f(x)$?
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Shift left $2$ units. Adding inside parentheses shifts left, not right.
Shift left $2$ units. Adding inside parentheses shifts left, not right.
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For $g(x)=f(x)-3$, how does the $y$-intercept change from $f(0)$?
For $g(x)=f(x)-3$, how does the $y$-intercept change from $f(0)$?
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It becomes $f(0)-3$. Subtracting $3$ from $f(0)$ gives the new intercept.
It becomes $f(0)-3$. Subtracting $3$ from $f(0)$ gives the new intercept.
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Which transformation occurs when $f(x)$ becomes $f(x-h)$?
Which transformation occurs when $f(x)$ becomes $f(x-h)$?
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Horizontal shift right $h$ units (left if $h<0$). Replaces $x$ with $x-h$ in the function.
Horizontal shift right $h$ units (left if $h<0$). Replaces $x$ with $x-h$ in the function.
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Which transformation occurs when $f(x)$ becomes $f(x)+k$?
Which transformation occurs when $f(x)$ becomes $f(x)+k$?
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Vertical shift up $k$ units (down if $k<0$). Adds constant to all output values.
Vertical shift up $k$ units (down if $k<0$). Adds constant to all output values.
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What key features must be labeled when sketching from a verbal description?
What key features must be labeled when sketching from a verbal description?
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Intercepts, extrema, increasing/decreasing, sign, end behavior, period. Essential elements for complete function analysis.
Intercepts, extrema, increasing/decreasing, sign, end behavior, period. Essential elements for complete function analysis.
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For $f(x)=(x-1)(x-5)$, on which interval is $f(x)$ negative?
For $f(x)=(x-1)(x-5)$, on which interval is $f(x)$ negative?
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$(1,5)$. Parabola opens upward, so it's negative between zeros.
$(1,5)$. Parabola opens upward, so it's negative between zeros.
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For $f(x)=(x-1)(x-5)$, what are the $x$-intercepts?
For $f(x)=(x-1)(x-5)$, what are the $x$-intercepts?
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$x=1$ and $x=5$. Set each factor to zero: $x-1=0$ or $x-5=0$.
$x=1$ and $x=5$. Set each factor to zero: $x-1=0$ or $x-5=0$.
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For $f(x)=x^2-4$, what is the $y$-intercept?
For $f(x)=x^2-4$, what is the $y$-intercept?
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$-4$. Substitute $x=0$: $f(0)=0^2-4=-4$.
$-4$. Substitute $x=0$: $f(0)=0^2-4=-4$.
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For $f(x)=x^2-4$, what are the $x$-intercepts?
For $f(x)=x^2-4$, what are the $x$-intercepts?
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$x=-2$ and $x=2$. Set $f(x)=0$: $x^2-4=0$, so $x=\pm 2$.
$x=-2$ and $x=2$. Set $f(x)=0$: $x^2-4=0$, so $x=\pm 2$.
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For $f(x)=x^2-4$, on which intervals is the function positive?
For $f(x)=x^2-4$, on which intervals is the function positive?
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$(-\infty,-2)\cup(2,\infty)$. Function is positive outside its zeros at $x=\pm 2$.
$(-\infty,-2)\cup(2,\infty)$. Function is positive outside its zeros at $x=\pm 2$.
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For $f(x)=x^2-4$, on which interval is the function negative?
For $f(x)=x^2-4$, on which interval is the function negative?
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$(-2,2)$. Function is negative between its zeros at $x=\pm 2$.
$(-2,2)$. Function is negative between its zeros at $x=\pm 2$.
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What is the meaning of the statement “$f(x)$ is decreasing on $(a,b)$”?
What is the meaning of the statement “$f(x)$ is decreasing on $(a,b)$”?
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For $a<x_1<x_2<b$, $f(x_1)>f(x_2)$. Larger inputs give smaller outputs throughout the interval.
For $a<x_1<x_2<b$, $f(x_1)>f(x_2)$. Larger inputs give smaller outputs throughout the interval.
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What is the meaning of the statement “$f(x)$ is increasing on $(a,b)$”?
What is the meaning of the statement “$f(x)$ is increasing on $(a,b)$”?
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For $a<x_1<x_2<b$, $f(x_1)<f(x_2)$. Larger inputs give larger outputs throughout the interval.
For $a<x_1<x_2<b$, $f(x_1)<f(x_2)$. Larger inputs give larger outputs throughout the interval.
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Identify an $x$-intercept from the table point $(-3,0)$.
Identify an $x$-intercept from the table point $(-3,0)$.
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$-3$. The $x$-intercept is the input value when output is zero.
$-3$. The $x$-intercept is the input value when output is zero.
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What does a table tell you about an $x$-intercept?
What does a table tell you about an $x$-intercept?
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It occurs where the output is $0$, at a point $(x,0)$. Find the row where output is zero.
It occurs where the output is $0$, at a point $(x,0)$. Find the row where output is zero.
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Identify the $y$-intercept from the table point $(0,7)$.
Identify the $y$-intercept from the table point $(0,7)$.
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$7$. The $y$-intercept is the output value when input is zero.
$7$. The $y$-intercept is the output value when input is zero.
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What does a table tell you about the $y$-intercept?
What does a table tell you about the $y$-intercept?
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It is the output when $x=0$, the ordered pair $(0,f(0))$. Find the row where input is zero.
It is the output when $x=0$, the ordered pair $(0,f(0))$. Find the row where input is zero.
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If $f(x)$ has period $4$, what is $f(x+12)$ equal to?
If $f(x)$ has period $4$, what is $f(x+12)$ equal to?
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$f(x)$. Since $12=3\times 4$, we get $f(x+12)=f(x)$.
$f(x)$. Since $12=3\times 4$, we get $f(x+12)=f(x)$.
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If $f(x)$ has period $6$, what is $f(x+6)$ equal to?
If $f(x)$ has period $6$, what is $f(x+6)$ equal to?
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$f(x)$. Since period is $6$, $f(x+6)=f(x)$ by definition.
$f(x)$. Since period is $6$, $f(x+6)=f(x)$ by definition.
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What is the definition of period $P$ for a periodic function?
What is the definition of period $P$ for a periodic function?
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The smallest $P>0$ such that $f(x+P)=f(x)$. Adding the period returns the same output.
The smallest $P>0$ such that $f(x+P)=f(x)$. Adding the period returns the same output.
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What is periodicity for a function?
What is periodicity for a function?
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A repeating pattern with some positive period $P$. Function values repeat at regular intervals.
A repeating pattern with some positive period $P$. Function values repeat at regular intervals.
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For $f(x)=-x^3$, what is the end behavior as $x\to\infty$ and $x\to-\infty$?
For $f(x)=-x^3$, what is the end behavior as $x\to\infty$ and $x\to-\infty$?
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As $x\to\infty$, $f(x)\to-\infty$; as $x\to-\infty$, $f(x)\to\infty$. Odd-degree polynomial with negative leading coefficient.
As $x\to\infty$, $f(x)\to-\infty$; as $x\to-\infty$, $f(x)\to\infty$. Odd-degree polynomial with negative leading coefficient.
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For $f(x)=x^3$, what is the end behavior as $x\to\infty$ and $x\to-\infty$?
For $f(x)=x^3$, what is the end behavior as $x\to\infty$ and $x\to-\infty$?
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As $x\to\infty$, $f(x)\to\infty$; as $x\to-\infty$, $f(x)\to-\infty$. Odd-degree polynomial with positive leading coefficient.
As $x\to\infty$, $f(x)\to\infty$; as $x\to-\infty$, $f(x)\to-\infty$. Odd-degree polynomial with positive leading coefficient.
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For $f(x)=-x^2$, what is the end behavior as $x\to\pm\infty$?
For $f(x)=-x^2$, what is the end behavior as $x\to\pm\infty$?
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$f(x)\to-\infty$ as $x\to\infty$ and as $x\to-\infty$. Downward-opening parabola goes to negative infinity both ways.
$f(x)\to-\infty$ as $x\to\infty$ and as $x\to-\infty$. Downward-opening parabola goes to negative infinity both ways.
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For $f(x)=x^2$, what is the end behavior as $x\to\pm\infty$?
For $f(x)=x^2$, what is the end behavior as $x\to\pm\infty$?
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$f(x)\to\infty$ as $x\to\infty$ and as $x\to-\infty$. Upward-opening parabola goes to infinity both ways.
$f(x)\to\infty$ as $x\to\infty$ and as $x\to-\infty$. Upward-opening parabola goes to infinity both ways.
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What is end behavior describing for a function’s graph?
What is end behavior describing for a function’s graph?
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What happens to $f(x)$ as $x\to\infty$ and $x\to-\infty$. Behavior as $x$ approaches positive and negative infinity.
What happens to $f(x)$ as $x\to\infty$ and $x\to-\infty$. Behavior as $x$ approaches positive and negative infinity.
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Which symmetry does $f(x)=x^3$ have: $y$-axis, origin, or none?
Which symmetry does $f(x)=x^3$ have: $y$-axis, origin, or none?
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Origin symmetry (odd). Since $f(-x)=(-x)^3=-x^3=-f(x)$, it's odd.
Origin symmetry (odd). Since $f(-x)=(-x)^3=-x^3=-f(x)$, it's odd.
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Which symmetry does $f(x)=x^2-1$ have: $y$-axis, origin, or none?
Which symmetry does $f(x)=x^2-1$ have: $y$-axis, origin, or none?
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$y$-axis symmetry (even). Since $f(-x)=(-x)^2-1=x^2-1=f(x)$, it's even.
$y$-axis symmetry (even). Since $f(-x)=(-x)^2-1=x^2-1=f(x)$, it's even.
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