Linear Functions - PSAT Math
Card 1 of 30
What is the standard form of a linear equation?
What is the standard form of a linear equation?
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$Ax+By=C$. General form where $A$, $B$, and $C$ are constants.
$Ax+By=C$. General form where $A$, $B$, and $C$ are constants.
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What is the point-slope form of a line with slope $m$ through $(x_1,y_1)$?
What is the point-slope form of a line with slope $m$ through $(x_1,y_1)$?
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$y-y_1=m(x-x_1)$. Uses a known point and slope to define the line.
$y-y_1=m(x-x_1)$. Uses a known point and slope to define the line.
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State the formula for slope between $(x_1,y_1)$ and $(x_2,y_2)$.
State the formula for slope between $(x_1,y_1)$ and $(x_2,y_2)$.
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$m=\frac{y_2-y_1}{x_2-x_1}$. Rise over run between two points.
$m=\frac{y_2-y_1}{x_2-x_1}$. Rise over run between two points.
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What do $m$ and $b$ represent in $y=mx+b$?
What do $m$ and $b$ represent in $y=mx+b$?
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$m$ is slope; $b$ is $y$-intercept. Slope determines steepness; $y$-intercept is where line crosses $y$-axis.
$m$ is slope; $b$ is $y$-intercept. Slope determines steepness; $y$-intercept is where line crosses $y$-axis.
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What is the slope-intercept form of a linear equation?
What is the slope-intercept form of a linear equation?
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$y=mx+b$. Standard form with slope $m$ and $y$-intercept $b$.
$y=mx+b$. Standard form with slope $m$ and $y$-intercept $b$.
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What is the slope of a horizontal line (written as $y=c$)?
What is the slope of a horizontal line (written as $y=c$)?
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$0$. Horizontal lines have no rise, so slope is zero.
$0$. Horizontal lines have no rise, so slope is zero.
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What is the slope of a vertical line (written as $x=c$)?
What is the slope of a vertical line (written as $x=c$)?
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Undefined. Vertical lines have infinite steepness (division by zero).
Undefined. Vertical lines have infinite steepness (division by zero).
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Identify the $x$-intercept in terms of $m$ and $b$ for $y=mx+b$.
Identify the $x$-intercept in terms of $m$ and $b$ for $y=mx+b$.
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$x=-\frac{b}{m}$ (if $m\ne 0$). Set $y=0$ and solve for $x$.
$x=-\frac{b}{m}$ (if $m\ne 0$). Set $y=0$ and solve for $x$.
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What is the slope of a line perpendicular to a line with slope $m$?
What is the slope of a line perpendicular to a line with slope $m$?
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$-\frac{1}{m}$ (if $m\ne 0$). Perpendicular slopes multiply to $-1$.
$-\frac{1}{m}$ (if $m\ne 0$). Perpendicular slopes multiply to $-1$.
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What is the slope of a line parallel to a line with slope $m$?
What is the slope of a line parallel to a line with slope $m$?
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$m$. Parallel lines have identical slopes.
$m$. Parallel lines have identical slopes.
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Find the slope of the line through $(2,5)$ and $(6,1)$.
Find the slope of the line through $(2,5)$ and $(6,1)$.
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$-1$. $m=\frac{1-5}{6-2}=\frac{-4}{4}=-1$.
$-1$. $m=\frac{1-5}{6-2}=\frac{-4}{4}=-1$.
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Find the equation in slope-intercept form of the line with slope $3$ and $y$-intercept $-2$.
Find the equation in slope-intercept form of the line with slope $3$ and $y$-intercept $-2$.
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$y=3x-2$. Direct substitution into $y=mx+b$ form.
$y=3x-2$. Direct substitution into $y=mx+b$ form.
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Find the equation in slope-intercept form through $(0,-4)$ and $(2,0)$.
Find the equation in slope-intercept form through $(0,-4)$ and $(2,0)$.
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$y=2x-4$. Slope is $\frac{0-(-4)}{2-0}=2$; $y$-intercept is $-4$.
$y=2x-4$. Slope is $\frac{0-(-4)}{2-0}=2$; $y$-intercept is $-4$.
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Find the $y$-intercept of the line $2x+3y=12$.
Find the $y$-intercept of the line $2x+3y=12$.
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$4$. Set $x=0$: $3y=12$, so $y=4$.
$4$. Set $x=0$: $3y=12$, so $y=4$.
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Find the slope of the line $5x-2y=10$.
Find the slope of the line $5x-2y=10$.
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$\frac{5}{2}$. Rewrite as $y=\frac{5}{2}x-5$; slope is coefficient of $x$.
$\frac{5}{2}$. Rewrite as $y=\frac{5}{2}x-5$; slope is coefficient of $x$.
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Find the equation of the line parallel to $y=-2x+7$ through $(3,1)$.
Find the equation of the line parallel to $y=-2x+7$ through $(3,1)$.
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$y=-2x+7$. Same slope $-2$; use point-slope: $y-1=-2(x-3)$.
$y=-2x+7$. Same slope $-2$; use point-slope: $y-1=-2(x-3)$.
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Find the equation of the line perpendicular to $y=\frac{1}{4}x-3$ through $(0,2)$.
Find the equation of the line perpendicular to $y=\frac{1}{4}x-3$ through $(0,2)$.
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$y=-4x+2$. Perpendicular slope is $-4$; passes through $(0,2)$.
$y=-4x+2$. Perpendicular slope is $-4$; passes through $(0,2)$.
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Identify whether $y=7x+1$ is increasing, decreasing, or constant.
Identify whether $y=7x+1$ is increasing, decreasing, or constant.
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Increasing. Positive slope ($7>0$) means function increases.
Increasing. Positive slope ($7>0$) means function increases.
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Find the function value $f(3)$ for $f(x)=-2x+9$.
Find the function value $f(3)$ for $f(x)=-2x+9$.
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$3$. $f(3)=-2(3)+9=-6+9=3$.
$3$. $f(3)=-2(3)+9=-6+9=3$.
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What is the relationship between slopes of parallel nonvertical lines?
What is the relationship between slopes of parallel nonvertical lines?
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They are equal: $m_1 = m_2$. Parallel lines have the same slope.
They are equal: $m_1 = m_2$. Parallel lines have the same slope.
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What is the $x$-intercept of $y = mx + b$ in terms of $m$ and $b$ (with $m \ne 0$)?
What is the $x$-intercept of $y = mx + b$ in terms of $m$ and $b$ (with $m \ne 0$)?
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$\left(-\frac{b}{m},0\right)$. Set $y=0$ and solve for $x$.
$\left(-\frac{b}{m},0\right)$. Set $y=0$ and solve for $x$.
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What is the $y$-intercept of the line $y = mx + b$?
What is the $y$-intercept of the line $y = mx + b$?
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$(0,b)$. Where the line crosses the y-axis (when $x=0$).
$(0,b)$. Where the line crosses the y-axis (when $x=0$).
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What is the equation of a vertical line through $x = a$?
What is the equation of a vertical line through $x = a$?
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$x = a$. Vertical lines have constant x-coordinate.
$x = a$. Vertical lines have constant x-coordinate.
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Find the $x$-intercept of $3x - 2y = 6$.
Find the $x$-intercept of $3x - 2y = 6$.
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$(2,0)$. Set $y=0$: $3x=6$, so $x=2$.
$(2,0)$. Set $y=0$: $3x=6$, so $x=2$.
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Find the slope of the line through $(2,7)$ and $(6,-1)$.
Find the slope of the line through $(2,7)$ and $(6,-1)$.
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$m=-2$. Apply slope formula: $m = \frac{-1-7}{6-2} = \frac{-8}{4} = -2$.
$m=-2$. Apply slope formula: $m = \frac{-1-7}{6-2} = \frac{-8}{4} = -2$.
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Find the equation in slope-intercept form of the line with slope $4$ and $y$-intercept $-2$.
Find the equation in slope-intercept form of the line with slope $4$ and $y$-intercept $-2$.
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$y = 4x - 2$. Substitute $m$ and $b$ into $y = mx + b$.
$y = 4x - 2$. Substitute $m$ and $b$ into $y = mx + b$.
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Find the equation of the line with slope $-\frac{1}{2}$ passing through $(4,3)$.
Find the equation of the line with slope $-\frac{1}{2}$ passing through $(4,3)$.
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$y = -\frac{1}{2}x + 5$. Use point-slope form: $y - 3 = -\frac{1}{2}(x - 4)$.
$y = -\frac{1}{2}x + 5$. Use point-slope form: $y - 3 = -\frac{1}{2}(x - 4)$.
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Find the $y$-intercept of $2x + 3y = 12$.
Find the $y$-intercept of $2x + 3y = 12$.
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$(0,4)$. Set $x=0$: $3y=12$, so $y=4$.
$(0,4)$. Set $x=0$: $3y=12$, so $y=4$.
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What is the slope of a line perpendicular to $y = \frac{2}{3}x - 1$?
What is the slope of a line perpendicular to $y = \frac{2}{3}x - 1$?
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$-\frac{3}{2}$. Negative reciprocal of $\frac{2}{3}$ is $-\frac{3}{2}$.
$-\frac{3}{2}$. Negative reciprocal of $\frac{2}{3}$ is $-\frac{3}{2}$.
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Evaluate the linear function $f(x)=2x-7$ at $x=5$.
Evaluate the linear function $f(x)=2x-7$ at $x=5$.
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$f(5)=3$. Substitute: $f(5) = 2(5) - 7 = 10 - 7 = 3$.
$f(5)=3$. Substitute: $f(5) = 2(5) - 7 = 10 - 7 = 3$.
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