GRE Quantitative Flashcards: Coordinate Geometry Graph Interpretation

What this deck covers

This deck focuses on Coordinate Geometry Graph Interpretation, giving you a quick way to review the definitions, rules, and examples that matter most for GRE Quantitative.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What is the vertex of the parabola $y=(x-2)^2-9$?

Answer: $(2,-9)$. Vertex form $y=a(x-h)^2+k$ identifies vertex at $(h,k)=(2,-9)$.

Flashcard 2: What is the slope formula for the line through points $(x_1,y_1)$ and $(x_2,y_2)$?

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.

Flashcard 3: Identify the axis of symmetry of the parabola $y=(x+5)^2+1$.

Answer: $x=-5$. In vertex form $y=(x-h)^2+k$, axis of symmetry is $x=h=-5$.

Flashcard 4: What does it mean graphically if a function is increasing on an interval?

Answer: As $x$ increases, $y$ increases (positive slope trend). The graph rises from left to right, reflecting a positive derivative.

Flashcard 5: Identify the $y$-intercept of the line $2x-3y=12$.

Answer: $(0,-4)$. Set $x=0$ in $2(0)-3y=12$ to solve $-3y=12$, so $y=-4$.

Flashcard 6: What is the condition on slopes $m_1$ and $m_2$ for two nonvertical lines to be perpendicular?

Answer: $m_1m_2=-1$. The product of slopes equals $-1$ for lines forming right angles.

Flashcard 7: What is the midpoint formula for the segment with endpoints $(x_1,y_1)$ and $(x_2,y_2)$?

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.

Flashcard 8: Which option is the slope of a line perpendicular to $y=-\frac{1}{4}x+7$?

Answer: $4$. Perpendicular slope is the negative reciprocal of $-\frac{1}{4}$, which is $4$.

Flashcard 9: Identify the distance between points $(0,0)$ and $(3,4)$.

Answer: $5$. Distance formula gives $\sqrt{(3-0)^2+(4-0)^2}=\sqrt{25}=5$.

Flashcard 10: Identify the slope of the line passing through $(2,3)$ and $(6,11)$.

Answer: $2$. Apply slope formula: $\frac{11-3}{6-2}=\frac{8}{4}=2$.

Flashcard 11: What is the slope-intercept form of a line with slope $m$ and $y$-intercept $b$?

Answer: $y=mx+b$. Describes the line with its slope and the point where it crosses the $y$-axis.

Flashcard 12: What is the slope of any vertical line in the coordinate plane?

Answer: Undefined. Infinite slope due to zero change in $x$, causing division by zero in the formula.

Flashcard 13: What is the equation of a line with slope $m$ passing through $(x_1,y_1)$ (point-slope form)?

Answer: $y-y_1=m(x-x_1)$. Expresses the linear relationship using a known point and the constant rate of change.

Flashcard 14: Identify the equation of the line parallel to $y=\frac{2}{3}x-1$ passing through $(0,5)$.

Answer: $y=\frac{2}{3}x+5$. Parallel lines share slope $\frac{2}{3}$, and $(0,5)$ gives $y$-intercept $5$.

Flashcard 15: What is the condition on slopes $m_1$ and $m_2$ for two nonvertical lines to be parallel?

Answer: $m_1=m_2$. Equal slopes ensure the lines maintain constant distance and never intersect.

Flashcard 16: What is the distance formula between points $(x_1,y_1)$ and $(x_2,y_2)$ in the coordinate plane?

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.

Flashcard 17: What does it mean graphically if a function is decreasing on an interval?

Answer: As $x$ increases, $y$ decreases (negative slope trend). The graph falls from left to right, reflecting a negative derivative.

Flashcard 18: What is the slope of any horizontal line in the coordinate plane?

Answer: $0$. No vertical change occurs as $x$ varies, resulting in zero rise over run.

Flashcard 19: Identify the $x$-intercept of the line $2x-3y=12$.

Answer: $(6,0)$. Set $y=0$ in $2x-3(0)=12$ to solve $2x=12$, so $x=6$.

Flashcard 20: What is the equation of the line with slope $3$ passing through $(1,-2)$?

Answer: $y=3x-5$. Point-slope form $y+2=3(x-1)$ simplifies to slope-intercept form.

Flashcard 21: Identify the midpoint of the segment with endpoints $(-4,7)$ and $(2,-1)$.

Answer: $(-1,3)$. Midpoint formula yields $\left(\frac{-4+2}{2}, \frac{7+(-1)}{2}\right)=(-1,3)$.

Flashcard 22: What are the $x$- and $y$-intercepts of the line in standard form $Ax+By=C$?

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 $y$.