Coordinate Geometry and Graph Interpretation - GRE Quantitative Reasoning
Card 1 of 22
What is the condition on slopes $m_1$ and $m_2$ for two nonvertical lines to be parallel?
What is the condition on slopes $m_1$ and $m_2$ for two nonvertical lines to be parallel?
Tap to reveal answer
$m_1=m_2$. Equal slopes ensure the lines maintain constant distance and never intersect.
$m_1=m_2$. Equal slopes ensure the lines maintain constant distance and never intersect.
← Didn't Know|Knew It →
Identify the axis of symmetry of the parabola $y=(x+5)^2+1$.
Identify the axis of symmetry of the parabola $y=(x+5)^2+1$.
Tap to reveal answer
$x=-5$. In vertex form $y=(x-h)^2+k$, axis of symmetry is $x=h=-5$.
$x=-5$. In vertex form $y=(x-h)^2+k$, axis of symmetry is $x=h=-5$.
← Didn't Know|Knew It →
What is the vertex of the parabola $y=(x-2)^2-9$?
What is the vertex of the parabola $y=(x-2)^2-9$?
Tap to reveal answer
$(2,-9)$. Vertex form $y=a(x-h)^2+k$ identifies vertex at $(h,k)=(2,-9)$.
$(2,-9)$. Vertex form $y=a(x-h)^2+k$ identifies vertex at $(h,k)=(2,-9)$.
← Didn't Know|Knew It →
Identify the equation of the line parallel to $y=\frac{2}{3}x-1$ passing through $(0,5)$.
Identify the equation of the line parallel to $y=\frac{2}{3}x-1$ passing through $(0,5)$.
Tap to reveal answer
$y=\frac{2}{3}x+5$. Parallel lines share slope $\frac{2}{3}$, and $(0,5)$ gives $y$-intercept $5$.
$y=\frac{2}{3}x+5$. Parallel lines share slope $\frac{2}{3}$, and $(0,5)$ gives $y$-intercept $5$.
← Didn't Know|Knew It →
Which option is the slope of a line perpendicular to $y=-\frac{1}{4}x+7$?
Which option is the slope of a line perpendicular to $y=-\frac{1}{4}x+7$?
Tap to reveal answer
$4$. Perpendicular slope is the negative reciprocal of $-\frac{1}{4}$, which is $4$.
$4$. Perpendicular slope is the negative reciprocal of $-\frac{1}{4}$, which is $4$.
← Didn't Know|Knew It →
Identify the $y$-intercept of the line $2x-3y=12$.
Identify the $y$-intercept of the line $2x-3y=12$.
Tap to reveal answer
$(0,-4)$. Set $x=0$ in $2(0)-3y=12$ to solve $-3y=12$, so $y=-4$.
$(0,-4)$. Set $x=0$ in $2(0)-3y=12$ to solve $-3y=12$, so $y=-4$.
← Didn't Know|Knew It →
Identify the $x$-intercept of the line $2x-3y=12$.
Identify the $x$-intercept of the line $2x-3y=12$.
Tap to reveal answer
$(6,0)$. Set $y=0$ in $2x-3(0)=12$ to solve $2x=12$, so $x=6$.
$(6,0)$. Set $y=0$ in $2x-3(0)=12$ to solve $2x=12$, so $x=6$.
← Didn't Know|Knew It →
What is the equation of the line with slope $3$ passing through $(1,-2)$?
What is the equation of the line with slope $3$ passing through $(1,-2)$?
Tap to reveal answer
$y=3x-5$. Point-slope form $y+2=3(x-1)$ simplifies to slope-intercept form.
$y=3x-5$. Point-slope form $y+2=3(x-1)$ simplifies to slope-intercept form.
← Didn't Know|Knew It →
Identify the distance between points $(0,0)$ and $(3,4)$.
Identify the distance between points $(0,0)$ and $(3,4)$.
Tap to reveal answer
$5$. Distance formula gives $\sqrt{(3-0)^2+(4-0)^2}=\sqrt{25}=5$.
$5$. Distance formula gives $\sqrt{(3-0)^2+(4-0)^2}=\sqrt{25}=5$.
← Didn't Know|Knew It →
Identify the midpoint of the segment with endpoints $(-4,7)$ and $(2,-1)$.
Identify the midpoint of the segment with endpoints $(-4,7)$ and $(2,-1)$.
Tap to reveal answer
$(-1,3)$. Midpoint formula yields $\left(\frac{-4+2}{2}, \frac{7+(-1)}{2}\right)=(-1,3)$.
$(-1,3)$. Midpoint formula yields $\left(\frac{-4+2}{2}, \frac{7+(-1)}{2}\right)=(-1,3)$.
← Didn't Know|Knew It →
What does it mean graphically if a function is decreasing on an interval?
What does it mean graphically if a function is decreasing on an interval?
Tap to reveal answer
As $x$ increases, $y$ decreases (negative slope trend). The graph falls from left to right, reflecting a negative derivative.
As $x$ increases, $y$ decreases (negative slope trend). The graph falls from left to right, reflecting a negative derivative.
← Didn't Know|Knew It →
What does it mean graphically if a function is increasing on an interval?
What does it mean graphically if a function is increasing on an interval?
Tap to reveal answer
As $x$ increases, $y$ increases (positive slope trend). The graph rises from left to right, reflecting a positive derivative.
As $x$ increases, $y$ increases (positive slope trend). The graph rises from left to right, reflecting a positive derivative.
← Didn't Know|Knew It →
What are the $x$- and $y$-intercepts of the line in standard form $Ax+By=C$?
What are the $x$- and $y$-intercepts of the line in standard form $Ax+By=C$?
Tap to reveal answer
$x\text{-int}=\frac{C}{A},\ y\text{-int}=\frac{C}{B}$. Solve for intercepts by setting $y=0$ for $x$ and $x=0$ for $y$.
$x\text{-int}=\frac{C}{A},\ y\text{-int}=\frac{C}{B}$. Solve for intercepts by setting $y=0$ for $x$ and $x=0$ for $y$.
← Didn't Know|Knew It →
What is the condition on slopes $m_1$ and $m_2$ for two nonvertical lines to be perpendicular?
What is the condition on slopes $m_1$ and $m_2$ for two nonvertical lines to be perpendicular?
Tap to reveal answer
$m_1m_2=-1$. The product of slopes equals $-1$ for lines forming right angles.
$m_1m_2=-1$. The product of slopes equals $-1$ for lines forming right angles.
← Didn't Know|Knew It →
What is the slope of any vertical line in the coordinate plane?
What is the slope of any vertical line in the coordinate plane?
Tap to reveal answer
Undefined. Infinite slope due to zero change in $x$, causing division by zero in the formula.
Undefined. Infinite slope due to zero change in $x$, causing division by zero in the formula.
← Didn't Know|Knew It →
What is the slope of any horizontal line in the coordinate plane?
What is the slope of any horizontal line in the coordinate plane?
Tap to reveal answer
$0$. No vertical change occurs as $x$ varies, resulting in zero rise over run.
$0$. No vertical change occurs as $x$ varies, resulting in zero rise over run.
← Didn't Know|Knew It →
What is the slope-intercept form of a line with slope $m$ and $y$-intercept $b$?
What is the slope-intercept form of a line with slope $m$ and $y$-intercept $b$?
Tap to reveal answer
$y=mx+b$. Describes the line with its slope and the point where it crosses the $y$-axis.
$y=mx+b$. Describes the line with its slope and the point where it crosses the $y$-axis.
← Didn't Know|Knew It →
What is the equation of a line with slope $m$ passing through $(x_1,y_1)$ (point-slope form)?
What is the equation of a line with slope $m$ passing through $(x_1,y_1)$ (point-slope form)?
Tap to reveal answer
$y-y_1=m(x-x_1)$. Expresses the linear relationship using a known point and the constant rate of change.
$y-y_1=m(x-x_1)$. Expresses the linear relationship using a known point and the constant rate of change.
← Didn't Know|Knew It →
What is the midpoint formula for the segment with endpoints $(x_1,y_1)$ and $(x_2,y_2)$?
What is the midpoint formula for the segment with endpoints $(x_1,y_1)$ and $(x_2,y_2)$?
Tap to reveal answer
$\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$. Averages the $x$- and $y$-coordinates to find the central point of the segment.
$\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$. Averages the $x$- and $y$-coordinates to find the central point of the segment.
← Didn't Know|Knew It →
What is the distance formula between points $(x_1,y_1)$ and $(x_2,y_2)$ in the coordinate plane?
What is the distance formula between points $(x_1,y_1)$ and $(x_2,y_2)$ in the coordinate plane?
Tap to reveal answer
$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Derives from the Pythagorean theorem applied to the horizontal and vertical distances between points.
$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Derives from the Pythagorean theorem applied to the horizontal and vertical distances between points.
← Didn't Know|Knew It →
What is the slope formula for the line through points $(x_1,y_1)$ and $(x_2,y_2)$?
What is the slope formula for the line through points $(x_1,y_1)$ and $(x_2,y_2)$?
Tap to reveal answer
$m=\frac{y_2-y_1}{x_2-x_1}$. Calculates the rate of change as the difference in $y$-coordinates divided by the difference in $x$-coordinates.
$m=\frac{y_2-y_1}{x_2-x_1}$. Calculates the rate of change as the difference in $y$-coordinates divided by the difference in $x$-coordinates.
← Didn't Know|Knew It →
Identify the slope of the line passing through $(2,3)$ and $(6,11)$.
Identify the slope of the line passing through $(2,3)$ and $(6,11)$.
Tap to reveal answer
$2$. Apply slope formula: $\frac{11-3}{6-2}=\frac{8}{4}=2$.
$2$. Apply slope formula: $\frac{11-3}{6-2}=\frac{8}{4}=2$.
← Didn't Know|Knew It →