Comparing Functions Represented in Different Ways - Algebra 2
Card 1 of 30
Which has the larger minimum: $f(x)=(x-1)^2-4$ or $g(x)=2(x+3)^2-1$?
Which has the larger minimum: $f(x)=(x-1)^2-4$ or $g(x)=2(x+3)^2-1$?
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$g$ has the larger minimum ($-1>-4$). Compare the $k$-values from vertex form; $-1>-4$.
$g$ has the larger minimum ($-1>-4$). Compare the $k$-values from vertex form; $-1>-4$.
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Identify the end behavior of $f(x)=-x^2+6x-1$ as $x\to\infty$.
Identify the end behavior of $f(x)=-x^2+6x-1$ as $x\to\infty$.
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$f(x)\to-\infty$. Negative leading coefficient causes the parabola to go down as $x$ increases.
$f(x)\to-\infty$. Negative leading coefficient causes the parabola to go down as $x$ increases.
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What does it mean for a function to be increasing on an interval?
What does it mean for a function to be increasing on an interval?
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As $x$ increases, $f(x)$ increases on that interval. The function's output values rise as the input values increase.
As $x$ increases, $f(x)$ increases on that interval. The function's output values rise as the input values increase.
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What does it mean for a function to be decreasing on an interval?
What does it mean for a function to be decreasing on an interval?
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As $x$ increases, $f(x)$ decreases on that interval. The function's output values fall as the input values increase.
As $x$ increases, $f(x)$ decreases on that interval. The function's output values fall as the input values increase.
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What is the range of a function in words?
What is the range of a function in words?
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The set of all possible output values (all $y$-values). All $y$-values that the function can produce as outputs.
The set of all possible output values (all $y$-values). All $y$-values that the function can produce as outputs.
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What is the domain of a function in words?
What is the domain of a function in words?
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The set of all allowed input values (all $x$-values). All $x$-values for which the function is defined.
The set of all allowed input values (all $x$-values). All $x$-values for which the function is defined.
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What is the vertex form of a quadratic function?
What is the vertex form of a quadratic function?
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$f(x)=a(x-h)^2+k$. Standard form that directly shows the vertex coordinates.
$f(x)=a(x-h)^2+k$. Standard form that directly shows the vertex coordinates.
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A quadratic has vertex $(2,7)$ and opens downward; what is its maximum value?
A quadratic has vertex $(2,7)$ and opens downward; what is its maximum value?
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Maximum value is $7$. For downward-opening parabolas, the vertex $y$-coordinate is the maximum.
Maximum value is $7$. For downward-opening parabolas, the vertex $y$-coordinate is the maximum.
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What transformation is represented by $-f(x)$ compared to $f(x)$?
What transformation is represented by $-f(x)$ compared to $f(x)$?
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Reflection across the $x$-axis. Negating the output flips the graph over the horizontal axis.
Reflection across the $x$-axis. Negating the output flips the graph over the horizontal axis.
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What transformation is represented by $f(x-c)$ compared to $f(x)$?
What transformation is represented by $f(x-c)$ compared to $f(x)$?
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Horizontal shift right $c$ units (left if $c<0$). Subtracting from the input moves the graph horizontally opposite direction.
Horizontal shift right $c$ units (left if $c<0$). Subtracting from the input moves the graph horizontally opposite direction.
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What transformation is represented by $f(x)+c$ compared to $f(x)$?
What transformation is represented by $f(x)+c$ compared to $f(x)$?
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Vertical shift up $c$ units (down if $c<0$). Adding to the output moves the graph vertically.
Vertical shift up $c$ units (down if $c<0$). Adding to the output moves the graph vertically.
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What is the slope between points $(x_1,y_1)$ and $(x_2,y_2)$?
What is the slope between points $(x_1,y_1)$ and $(x_2,y_2)$?
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$m=
rac{y_2-y_1}{x_2-x_1}$. Rise over run; the change in $y$ divided by the change in $x$.
$m= rac{y_2-y_1}{x_2-x_1}$. Rise over run; the change in $y$ divided by the change in $x$.
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If a line has equation $y=mx+b$, what do $m$ and $b$ represent?
If a line has equation $y=mx+b$, what do $m$ and $b$ represent?
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$m$ is slope; $b$ is $y$-intercept. Slope-intercept form where $m$ determines steepness and $b$ is the starting value.
$m$ is slope; $b$ is $y$-intercept. Slope-intercept form where $m$ determines steepness and $b$ is the starting value.
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For $f(x)=a(x-h)^2+k$, when does the quadratic have a minimum value?
For $f(x)=a(x-h)^2+k$, when does the quadratic have a minimum value?
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When $a>0$, minimum value is $k$. When the parabola opens upward, the vertex gives the lowest point.
When $a>0$, minimum value is $k$. When the parabola opens upward, the vertex gives the lowest point.
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A table shows $f(0)=2$ and $f(3)=11$; find the average rate of change on $[0,3]$.
A table shows $f(0)=2$ and $f(3)=11$; find the average rate of change on $[0,3]$.
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$\frac{11-2}{3-0}=3$. Use the average rate formula with the table values over the interval.
$\frac{11-2}{3-0}=3$. Use the average rate formula with the table values over the interval.
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Find the slope of the line through $(1,2)$ and $(5,10)$.
Find the slope of the line through $(1,2)$ and $(5,10)$.
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$m= \frac{10-2}{5-1}=2$. Use the slope formula with the two given points.
$m= \frac{10-2}{5-1}=2$. Use the slope formula with the two given points.
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Identify the vertex of $f(x)=2(x-3)^2-5$.
Identify the vertex of $f(x)=2(x-3)^2-5$.
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$(3,-5)$. Read the vertex coordinates directly from the vertex form.
$(3,-5)$. Read the vertex coordinates directly from the vertex form.
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If a line has equation $y=mx+b$, what do $m$ and $b$ represent?
If a line has equation $y=mx+b$, what do $m$ and $b$ represent?
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$m$ is slope; $b$ is $y$-intercept. Slope-intercept form where $m$ determines steepness and $b$ is the starting value.
$m$ is slope; $b$ is $y$-intercept. Slope-intercept form where $m$ determines steepness and $b$ is the starting value.
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For $f(x)=a(x-h)^2+k$, when does the quadratic have a minimum value?
For $f(x)=a(x-h)^2+k$, when does the quadratic have a minimum value?
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When $a>0$, minimum value is $k$. When the parabola opens upward, the vertex gives the lowest point.
When $a>0$, minimum value is $k$. When the parabola opens upward, the vertex gives the lowest point.
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Which is larger: maximum of $f(x)=-2(x-1)^2+3$ or maximum of $g(x)=-(x+2)^2+5$?
Which is larger: maximum of $f(x)=-2(x-1)^2+3$ or maximum of $g(x)=-(x+2)^2+5$?
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$g$ has the larger maximum ($5 > 3$). Compare the $k$-values from vertex form; $5 > 3$.
$g$ has the larger maximum ($5 > 3$). Compare the $k$-values from vertex form; $5 > 3$.
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What is the domain of a function in words?
What is the domain of a function in words?
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The set of all allowed input values (all $x$-values). All $x$-values for which the function is defined.
The set of all allowed input values (all $x$-values). All $x$-values for which the function is defined.
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Identify the minimum value of $f(x)=
rac{1}{2}(x-6)^2-2$.
Identify the minimum value of $f(x)= rac{1}{2}(x-6)^2-2$.
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Minimum value is $-2$. Positive $a$ means upward opening, so vertex gives minimum at $y=k$.
Minimum value is $-2$. Positive $a$ means upward opening, so vertex gives minimum at $y=k$.
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A function is described as "starts at $y=3$ when $x=0$ and rises $2$ per $1$ right"; what is its equation?
A function is described as "starts at $y=3$ when $x=0$ and rises $2$ per $1$ right"; what is its equation?
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$y=2x+3$. Linear function with slope $2$ and $y$-intercept $3$.
$y=2x+3$. Linear function with slope $2$ and $y$-intercept $3$.
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Compute the average rate of change of $f$ from $x=1$ to $x=4$ if $f(1)=3$ and $f(4)=15$.
Compute the average rate of change of $f$ from $x=1$ to $x=4$ if $f(1)=3$ and $f(4)=15$.
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$
rac{15-3}{4-1}=4$. Apply the average rate of change formula with the given values.
$ rac{15-3}{4-1}=4$. Apply the average rate of change formula with the given values.
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If $f(x)$ is shifted to $g(x)=f(x)+5$, how do their maximum values compare?
If $f(x)$ is shifted to $g(x)=f(x)+5$, how do their maximum values compare?
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Maximum of $g$ is $5$ more than maximum of $f$. Vertical shifts add the same amount to all function values.
Maximum of $g$ is $5$ more than maximum of $f$. Vertical shifts add the same amount to all function values.
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A quadratic has vertex $(2,7)$ and opens downward; what is its maximum value?
A quadratic has vertex $(2,7)$ and opens downward; what is its maximum value?
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Maximum value is $7$. For downward-opening parabolas, the vertex $y$-coordinate is the maximum.
Maximum value is $7$. For downward-opening parabolas, the vertex $y$-coordinate is the maximum.
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A function increases from $x=1$ to $x=5$; which is larger, $f(1)$ or $f(5)$?
A function increases from $x=1$ to $x=5$; which is larger, $f(1)$ or $f(5)$?
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$f(5)$ is larger. On increasing intervals, larger $x$-values produce larger function values.
$f(5)$ is larger. On increasing intervals, larger $x$-values produce larger function values.
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A function decreases from $x=-2$ to $x=4$; which is larger, $f(-2)$ or $f(4)$?
A function decreases from $x=-2$ to $x=4$; which is larger, $f(-2)$ or $f(4)$?
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$f(-2)$ is larger. On decreasing intervals, smaller $x$-values produce larger function values.
$f(-2)$ is larger. On decreasing intervals, smaller $x$-values produce larger function values.
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Identify the range of $f(x)=(x-2)^2+5$.
Identify the range of $f(x)=(x-2)^2+5$.
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Range: $y\ge^5$. Upward-opening parabola with vertex at $y=5$ gives range $y \geq 5$.
Range: $y\ge^5$. Upward-opening parabola with vertex at $y=5$ gives range $y \geq 5$.
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If $f(x)=x^2$ and $g(x)=(x-4)^2$, how do their minimum values compare?
If $f(x)=x^2$ and $g(x)=(x-4)^2$, how do their minimum values compare?
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They are equal; both minima are $0$. Horizontal shifts don't change the minimum value, only its location.
They are equal; both minima are $0$. Horizontal shifts don't change the minimum value, only its location.
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