Linear Functions - SAT Math
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Equation of a line in slope-intercept form.
Equation of a line in slope-intercept form.
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$y = mx + b$.
$y = mx + b$.
Equation of a line with slope $m$ through $(x_1, y_1)$.
Equation of a line with slope $m$ through $(x_1, y_1)$.
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$y - y_1 = m(x - x_1)$.
$y - y_1 = m(x - x_1)$.
Find equation of line passing through (0, 4) with slope -2.
Find equation of line passing through (0, 4) with slope -2.
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$y = -2x + 4.$
$y = -2x + 4.$
Find equation of line with slope 4 passing through (2, 3).
Find equation of line with slope 4 passing through (2, 3).
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$y - 3 = 4(x - 2) ⇒ y = 4x - 5.$
$y - 3 = 4(x - 2) ⇒ y = 4x - 5.$
Find equation of line with x-intercept 6 and y-intercept -3.
Find equation of line with x-intercept 6 and y-intercept -3.
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y = (-3/-6)x - 3 ⇒ $y=0.5x - 3$
y = (-3/-6)x - 3 ⇒ $y=0.5x - 3$
Find slope between (3, 4) and (7, 8).
Find slope between (3, 4) and (7, 8).
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$(8 - 4)/(7 - 3) = 4/4 = 1.$
$(8 - 4)/(7 - 3) = 4/4 = 1.$
Find slope of line through (-1, 3) and (-1, -2).
Find slope of line through (-1, 3) and (-1, -2).
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Undefined (vertical line).
Undefined (vertical line).
Find slope of line through (-2, 5) and (4, 5).
Find slope of line through (-2, 5) and (4, 5).
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$(5 - 5)/(4 - (-2)) = 0 ⇒ horizontal line.$
$(5 - 5)/(4 - (-2)) = 0 ⇒ horizontal line.$
Find slope of line y = -3x + 12.
Find slope of line y = -3x + 12.
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-3
-3
Find the slope of a line parallel to y = -3x + 9.
Find the slope of a line parallel to y = -3x + 9.
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-3
-3
Find the y-intercept of $y = 3x^2 - 5x + 2$.
Find the y-intercept of $y = 3x^2 - 5x + 2$.
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Plug in $x = 0$: $y = 2$.
Plug in $x = 0$: $y = 2$.
Find y-intercept of y = 5x - 9.
Find y-intercept of y = 5x - 9.
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-9
-9
For $y = |x|$, describe the rate of change.
For $y = |x|$, describe the rate of change.
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Slope = 1 for $x > 0$, slope = -1 for $x < 0$.
Slope = 1 for $x > 0$, slope = -1 for $x < 0$.
Formula for a linear function.
Formula for a linear function.
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$y = mx + b$.
$y = mx + b$.
Formula for slope between $(x_1, y_1)$ and $(x_2, y_2)$.
Formula for slope between $(x_1, y_1)$ and $(x_2, y_2)$.
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$m = \frac{y_2 - y_1}{x_2 - x_1}$.
$m = \frac{y_2 - y_1}{x_2 - x_1}$.
Given 2x + 3y = 12, find y-intercept.
Given 2x + 3y = 12, find y-intercept.
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Set $x=0$ ⇒ 3y = 12 ⇒ y = 4 ⇒ (0, 4).
Set $x=0$ ⇒ 3y = 12 ⇒ y = 4 ⇒ (0, 4).
Given 3x - 6y = 12, find slope.
Given 3x - 6y = 12, find slope.
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$-A/B = -3/(-6) = \tfrac{1}{2}.$
$-A/B = -3/(-6) = \tfrac{1}{2}.$
How can you check whether (x, y) is a solution to a linear equation?
How can you check whether (x, y) is a solution to a linear equation?
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Substitute x and y into the equation and see if it makes a true statement.
Substitute x and y into the equation and see if it makes a true statement.
How do you find y-intercept of a line given slope and a point?
How do you find y-intercept of a line given slope and a point?
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Substitute the point (x, y) and slope m into $y=mx+b$ to solve for b.
Substitute the point (x, y) and slope m into $y=mx+b$ to solve for b.
If $y = 5x + 2$, find the rate of change and initial value.
If $y = 5x + 2$, find the rate of change and initial value.
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Rate = 5, initial value = 2.
Rate = 5, initial value = 2.
If $y = mx + b$, what happens if $m$ is negative?
If $y = mx + b$, what happens if $m$ is negative?
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The quantity decreases linearly.
The quantity decreases linearly.
If line passes through (1, 5) and (5, 1), find its slope.
If line passes through (1, 5) and (5, 1), find its slope.
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$(1 - 5)/(5 - 1) = -4/4 = -1.$
$(1 - 5)/(5 - 1) = -4/4 = -1.$
If two points have the same x-coordinate, what type of line is it?
If two points have the same x-coordinate, what type of line is it?
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Vertical (undefined slope).
Vertical (undefined slope).
If two points have the same y-coordinate, what type of line is it?
If two points have the same y-coordinate, what type of line is it?
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Horizontal (zero slope).
Horizontal (zero slope).
If y = 7 when $x=0$, and y = 3 when x = 2, find slope.
If y = 7 when $x=0$, and y = 3 when x = 2, find slope.
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$(3 - 7)/(2 - 0) = -4/2 = -2.$
$(3 - 7)/(2 - 0) = -4/2 = -2.$
If y increases by 8 when x increases by 4, what is the slope?
If y increases by 8 when x increases by 4, what is the slope?
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$m=\frac{8}{4}=2$.
$m=\frac{8}{4}=2$.
In $y = mx + b$, what does $b$ represent?
In $y = mx + b$, what does $b$ represent?
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The initial value or $y$-intercept.
The initial value or $y$-intercept.
In $y = mx + b$, what does $m$ represent?
In $y = mx + b$, what does $m$ represent?
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The rate of change or slope (change in $y$ per unit change in $x$).
The rate of change or slope (change in $y$ per unit change in $x$).
Line has slope 1/3 and passes through (0, -6). Write equation.
Line has slope 1/3 and passes through (0, -6). Write equation.
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$y = (1/3)x - 6.$
$y = (1/3)x - 6.$
Slope of a line parallel to one with slope $m$.
Slope of a line parallel to one with slope $m$.
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$m$ (same slope).
$m$ (same slope).
Slope of a line perpendicular to one with slope $m$.
Slope of a line perpendicular to one with slope $m$.
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$-\tfrac{1}{m}$.
$-\tfrac{1}{m}$.
What does it mean if slope $m$ is negative in $y = mx + b$?
What does it mean if slope $m$ is negative in $y = mx + b$?
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The quantity decreases at a constant rate.
The quantity decreases at a constant rate.
What does it mean if two linear equations have infinitely many solutions?
What does it mean if two linear equations have infinitely many solutions?
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They represent the same line (same slope and y-intercept).
They represent the same line (same slope and y-intercept).
What does it mean if two linear equations have no solution?
What does it mean if two linear equations have no solution?
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Their lines are parallel and distinct (same slope, different intercepts).
Their lines are parallel and distinct (same slope, different intercepts).
What does the slope m = 0 represent?
What does the slope m = 0 represent?
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A horizontal line; no change in y as x changes.
A horizontal line; no change in y as x changes.
What does the slope of a linear function represent?
What does the slope of a linear function represent?
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The rate of change — constant for all $x$.
The rate of change — constant for all $x$.
What is the formula for slope (m) between two points $(x_1, y_1)$ and $(x_2, y_2)$?
What is the formula for slope (m) between two points $(x_1, y_1)$ and $(x_2, y_2)$?
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$m=\frac{y_2-y_1}{x_2-x_1}$.
$m=\frac{y_2-y_1}{x_2-x_1}$.
What is the meaning of slope in a linear model?
What is the meaning of slope in a linear model?
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Rate of change of y with respect to x; how much y increases when x increases by 1.
Rate of change of y with respect to x; how much y increases when x increases by 1.
What is the meaning of y-intercept in a linear model?
What is the meaning of y-intercept in a linear model?
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The value of y when $x=0$.
The value of y when $x=0$.
What is the point-slope form of a line?
What is the point-slope form of a line?
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$y-y_1=m(x-x_1)$.
$y-y_1=m(x-x_1)$.
What is the slope of a line that goes through (0, 5) and (10, 0)?
What is the slope of a line that goes through (0, 5) and (10, 0)?
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$(0 - 5)/(10 - 0) = -\tfrac{1}{2}.$
$(0 - 5)/(10 - 0) = -\tfrac{1}{2}.$
What is the slope-intercept form of a line?
What is the slope-intercept form of a line?
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$y=mx+b$.
$y=mx+b$.
What kind of line has an undefined slope?
What kind of line has an undefined slope?
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A vertical line ($x=\text{constant}$).
A vertical line ($x=\text{constant}$).
What’s the algebraic goal when solving a linear equation?
What’s the algebraic goal when solving a linear equation?
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Isolate the variable on one side of the equation.
Isolate the variable on one side of the equation.