Compare Functions in Different Representations - 8th Grade Math
Card 1 of 30
Which statement best compares slopes: $m=-2$ versus $m=1$?
Which statement best compares slopes: $m=-2$ versus $m=1$?
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$1$ is greater than $-2$. Positive slopes indicate increasing functions; negative slopes indicate decreasing.
$1$ is greater than $-2$. Positive slopes indicate increasing functions; negative slopes indicate decreasing.
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What is the slope of a line passing through $(0,3)$ and $(2,7)$?
What is the slope of a line passing through $(0,3)$ and $(2,7)$?
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$2$. Using slope formula: $ rac{7-3}{2-0}= rac{4}{2}=2$.
$2$. Using slope formula: $ rac{7-3}{2-0}= rac{4}{2}=2$.
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What property of a linear function equals its rate of change in $y=mx+b$?
What property of a linear function equals its rate of change in $y=mx+b$?
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The slope $m$. In $y=mx+b$, the coefficient $m$ represents how much $y$ changes per unit change in $x$.
The slope $m$. In $y=mx+b$, the coefficient $m$ represents how much $y$ changes per unit change in $x$.
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What is the rate of change of the function $y=-3x+7$?
What is the rate of change of the function $y=-3x+7$?
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$-3$. The coefficient of $x$ in $y=mx+b$ form gives the rate of change.
$-3$. The coefficient of $x$ in $y=mx+b$ form gives the rate of change.
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What is the initial value (starting value) of the function $y=4x-9$?
What is the initial value (starting value) of the function $y=4x-9$?
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$-9$. The initial value is the $y$-intercept, found when $x=0$: $y=4(0)-9=-9$.
$-9$. The initial value is the $y$-intercept, found when $x=0$: $y=4(0)-9=-9$.
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Which function has the greater rate of change: $f(x)=2x+1$ or $g(x)=-x+5$?
Which function has the greater rate of change: $f(x)=2x+1$ or $g(x)=-x+5$?
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$f(x)$. $f(x)$ has slope $2$ while $g(x)$ has slope $-1$, so $f(x)$ changes faster.
$f(x)$. $f(x)$ has slope $2$ while $g(x)$ has slope $-1$, so $f(x)$ changes faster.
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Which function is increasing: $f(x)=-
rac{1}{2}x+3$ or $g(x)=
rac{3}{4}x-2$?
Which function is increasing: $f(x)=- rac{1}{2}x+3$ or $g(x)= rac{3}{4}x-2$?
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$g(x)$. $g(x)$ has positive slope $ rac{3}{4}$, while $f(x)$ has negative slope $- rac{1}{2}$.
$g(x)$. $g(x)$ has positive slope $ rac{3}{4}$, while $f(x)$ has negative slope $- rac{1}{2}$.
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Identify the rate of change for a function described as "$y$ decreases $5$ for each increase of $1$ in $x$".
Identify the rate of change for a function described as "$y$ decreases $5$ for each increase of $1$ in $x$".
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$-5$. A decrease of $5$ per unit increase means the slope is negative: $-5$.
$-5$. A decrease of $5$ per unit increase means the slope is negative: $-5$.
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Identify the initial value for a function described as "When $x=0$, $y=12$".
Identify the initial value for a function described as "When $x=0$, $y=12$".
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$12$. The initial value is the $y$-value when $x=0$.
$12$. The initial value is the $y$-value when $x=0$.
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Which representation shows a constant rate of change: a straight-line graph or a curved graph?
Which representation shows a constant rate of change: a straight-line graph or a curved graph?
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A straight-line graph. Linear functions have constant slope, appearing as straight lines on graphs.
A straight-line graph. Linear functions have constant slope, appearing as straight lines on graphs.
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What formula finds rate of change between points $(x_1,y_1)$ and $(x_2,y_2)$?
What formula finds rate of change between points $(x_1,y_1)$ and $(x_2,y_2)$?
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$\frac{y_2-y_1}{x_2-x_1}$. This formula calculates slope as rise over run between two points.
$\frac{y_2-y_1}{x_2-x_1}$. This formula calculates slope as rise over run between two points.
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Which table shows the greater rate of change: A has $(0,1),(2,5)$; B has $(0,1),(2,3)$?
Which table shows the greater rate of change: A has $(0,1),(2,5)$; B has $(0,1),(2,3)$?
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Table A. Table A: slope $= rac{5-1}{2-0}=2$; Table B: slope $= rac{3-1}{2-0}=1$.
Table A. Table A: slope $= rac{5-1}{2-0}=2$; Table B: slope $= rac{3-1}{2-0}=1$.
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What is the slope of a horizontal line (constant function) such as $y=6$?
What is the slope of a horizontal line (constant function) such as $y=6$?
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$0$. Horizontal lines have no vertical change, so slope equals zero.
$0$. Horizontal lines have no vertical change, so slope equals zero.
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What is the slope of a vertical line such as $x=2$?
What is the slope of a vertical line such as $x=2$?
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Undefined. Vertical lines have no horizontal change, making slope division by zero.
Undefined. Vertical lines have no horizontal change, making slope division by zero.
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Which function has the greater initial value: $f(x)=3x-4$ or $g(x)=-2x+1$?
Which function has the greater initial value: $f(x)=3x-4$ or $g(x)=-2x+1$?
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$g(x)$. $f(0)=-4$ and $g(0)=1$, so $g(x)$ has the greater initial value.
$g(x)$. $f(0)=-4$ and $g(0)=1$, so $g(x)$ has the greater initial value.
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Which function decreases faster: $f(x)=-4x+2$ or $g(x)=-x-10$?
Which function decreases faster: $f(x)=-4x+2$ or $g(x)=-x-10$?
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$f(x)$. $f(x)$ has slope $-4$ (more negative) versus $g(x)$ with slope $-1$.
$f(x)$. $f(x)$ has slope $-4$ (more negative) versus $g(x)$ with slope $-1$.
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Identify the missing value if a linear table has constant rate $3$: $(0,2)$, $(1,5)$, $(2,?)$.
Identify the missing value if a linear table has constant rate $3$: $(0,2)$, $(1,5)$, $(2,?)$.
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$8$. With rate $3$, each $x$ increase of $1$ adds $3$ to $y$: $5+3=8$.
$8$. With rate $3$, each $x$ increase of $1$ adds $3$ to $y$: $5+3=8$.
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What is the rate of change from the table points $ (1,5) $ and $ (3,9) $?
What is the rate of change from the table points $ (1,5) $ and $ (3,9) $?
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$2$. Using slope formula: $\frac{9-5}{3-1}=\frac{4}{2}=2$.
$2$. Using slope formula: $\frac{9-5}{3-1}=\frac{4}{2}=2$.
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Which function has greater slope: line through $(0,0)$ and $(4,8)$ or $y=
rac{3}{2}x+1$?
Which function has greater slope: line through $(0,0)$ and $(4,8)$ or $y= rac{3}{2}x+1$?
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The line through $(0,0)$ and $(4,8)$. Line has slope $ rac{8-0}{4-0}=2$, greater than $ rac{3}{2}$ from the equation.
The line through $(0,0)$ and $(4,8)$. Line has slope $ rac{8-0}{4-0}=2$, greater than $ rac{3}{2}$ from the equation.
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Which is larger: the rate of change $\frac{1}{3}$ or $\frac{2}{5}$?
Which is larger: the rate of change $\frac{1}{3}$ or $\frac{2}{5}$?
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$\frac{2}{5}$. Converting to decimals: $\frac{1}{3} \approx 0.33$ and $\frac{2}{5} = 0.4$.
$\frac{2}{5}$. Converting to decimals: $\frac{1}{3} \approx 0.33$ and $\frac{2}{5} = 0.4$.
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What is the slope of a vertical line given by $x=-2$?
What is the slope of a vertical line given by $x=-2$?
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Undefined. Vertical lines have zero run in the slope formula.
Undefined. Vertical lines have zero run in the slope formula.
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Identify whether the function is increasing, decreasing, or constant if $m<0$.
Identify whether the function is increasing, decreasing, or constant if $m<0$.
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Decreasing. Negative slopes go down from left to right.
Decreasing. Negative slopes go down from left to right.
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Identify whether the function is increasing, decreasing, or constant if $m>0$.
Identify whether the function is increasing, decreasing, or constant if $m>0$.
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Increasing. Positive slopes go up from left to right.
Increasing. Positive slopes go up from left to right.
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What is the rate of change from the table points $(1,4)$ and $(3,10)$?
What is the rate of change from the table points $(1,4)$ and $(3,10)$?
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$3$. Use $m=\frac{10-4}{3-1}=\frac{6}{2}=3$.
$3$. Use $m=\frac{10-4}{3-1}=\frac{6}{2}=3$.
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What is the rate of change from the table points $(0,-1)$ and $(2,5)$?
What is the rate of change from the table points $(0,-1)$ and $(2,5)$?
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$3$. Use $m=\frac{5-(-1)}{2-0}=\frac{6}{2}=3$.
$3$. Use $m=\frac{5-(-1)}{2-0}=\frac{6}{2}=3$.
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Which function has the greater rate of change: $f(x)=2x-1$ or table $(0,0)$, $(2,3)$?
Which function has the greater rate of change: $f(x)=2x-1$ or table $(0,0)$, $(2,3)$?
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$f(x)=2x-1$. $f$ has slope $2$; table has slope $\frac{3-0}{2-0}=1.5$.
$f(x)=2x-1$. $f$ has slope $2$; table has slope $\frac{3-0}{2-0}=1.5$.
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Which function has the greater rate of change: $g(x)=-x+6$ or table $(1,2)$, $(4,8)$?
Which function has the greater rate of change: $g(x)=-x+6$ or table $(1,2)$, $(4,8)$?
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The table function. $g$ has slope $-1$; table has slope $\frac{8-2}{4-1}=2$.
The table function. $g$ has slope $-1$; table has slope $\frac{8-2}{4-1}=2$.
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Which function is steeper: $y=\frac{1}{2}x+1$ or $y=-\frac{3}{4}x+2$?
Which function is steeper: $y=\frac{1}{2}x+1$ or $y=-\frac{3}{4}x+2$?
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$y=-\frac{3}{4}x+2$. Compare absolute values: $|\frac{1}{2}|<|\frac{3}{4}|$.
$y=-\frac{3}{4}x+2$. Compare absolute values: $|\frac{1}{2}|<|\frac{3}{4}|$.
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Which slope is greater: $m_1=-2$ or $m_2=-\frac{1}{2}$?
Which slope is greater: $m_1=-2$ or $m_2=-\frac{1}{2}$?
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$m_2=-\frac{1}{2}$. $-\frac{1}{2}$ is closer to zero, so it's greater.
$m_2=-\frac{1}{2}$. $-\frac{1}{2}$ is closer to zero, so it's greater.
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Which function has the greater initial value: $f(x)=3x-2$ or $g(x)=3x+5$?
Which function has the greater initial value: $f(x)=3x-2$ or $g(x)=3x+5$?
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$g(x)=3x+5$. Compare y-intercepts: $-2<5$.
$g(x)=3x+5$. Compare y-intercepts: $-2<5$.
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