PSAT Math Flashcards: Graphs

What this deck covers

This deck focuses on Graphs, giving you a quick way to review the definitions, rules, and examples that matter most for PSAT Math.

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 distance between $(1,-2)$ and $(4,2)$?

Answer: $5$. Apply distance formula: $\sqrt{(4-1)^2+(2-(-2))^2}=\sqrt{9+16}$.

Flashcard 2: What is the vertex form of a quadratic function?

Answer: $y=a(x-h)^2+k$. Shows parabola with vertex at $(h,k)$ and vertical stretch $a$.

Flashcard 3: Identify the transformation of $y=-f(x)$ relative to $y=f(x)$.

Answer: Reflect across the $x$-axis. Negative sign flips all $y$-values across horizontal axis.

Flashcard 4: What is the equation of a line with slope $3$ and $y$-intercept $-4$?

Answer: $y=3x-4$. Substitute $m$ and $b$ into $y=mx+b$.

Flashcard 5: What is the slope of a line parallel to $y=-\frac{1}{3}x+7$?

Answer: $m=-\frac{1}{3}$. Parallel lines have identical slopes.

Flashcard 6: State the distance formula between points $(x_1,y_1)$ and $(x_2,y_2)$.

Answer: $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Apply the Pythagorean theorem to find diagonal distance.

Flashcard 7: What is the slope of a vertical line (for example, $x=-4$)?

Answer: Slope is undefined. Vertical lines have infinite rise over zero run.

Flashcard 8: What is the slope of a horizontal line (for example, $y=7$)?

Answer: $m=0$. Horizontal lines have no rise, so slope equals zero.

Flashcard 9: What is the slope of a line passing through $(2,5)$ and $(6,1)$?

Answer: $-1$. Using slope formula: $\frac{1-5}{6-2}=\frac{-4}{4}=-1$.

Flashcard 10: What is the equation of the vertical line passing through $(4,0)$?

Answer: $x=4$. All vertical lines have form $x=k$ for constant $k$.

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

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

Flashcard 12: What is the average rate of change of $f(x)=x^2$ from $x=1$ to $x=3$?

Answer: $4$. $\frac{f(3)-f(1)}{3-1}=\frac{9-1}{2}=4$.

Flashcard 13: State the point-slope form of a line with slope $m$ through $(x_1,y_1)$.

Answer: $y-y_1=m(x-x_1)$. Expresses a line using a known point and the slope.

Flashcard 14: Find the distance between $(1,2)$ and $(4,6)$.

Answer: $5$. $d=\sqrt{(4-1)^2+(6-2)^2}=\sqrt{9+16}=\sqrt{25}=5$.

Flashcard 15: Identify the $x$-intercept of the line $y=-2x+8$.

Answer: $(4,0)$. Set $y=0$: $0=-2x+8$, so $x=4$.

Flashcard 16: What is the midpoint formula for the segment with endpoints $(x_1,y_1)$ and $(x_2,y_2)$?

Answer: $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$. Average the x-coordinates and y-coordinates separately.

Flashcard 17: What is the vertex of $y=(x-4)^2+1$?

Answer: $(4,1)$. Vertex form $(x-h)^2+k$ has vertex at $(h,k)$.

Flashcard 18: What is the slope of a line parallel to $y=-3x+7$?

Answer: $-3$. Parallel lines have identical slopes.

Flashcard 19: Find the midpoint of the segment with endpoints $(2,-1)$ and $(8,5)$.

Answer: $(5,2)$. $\left(\frac{2+8}{2},\frac{-1+5}{2}\right)=(5,2)$.

Flashcard 20: Find the slope of the line through $(2,5)$ and $(6,1)$.

Answer: $m=-1$. Using slope formula: $\frac{1-5}{6-2}=\frac{-4}{4}=-1$.

Flashcard 21: Identify the equation of the line through $(1,2)$ and $(3,6)$ in slope-intercept form.

Answer: $y=2x$. Slope is $\frac{6-2}{3-1}=2$; passes through origin.

Flashcard 22: Identify the $x$-intercept of $y=2x-8$.

Answer: $(4,0)$. Set $y=0$: $0=2x-8$, so $x=4$.

Flashcard 23: What is the slope formula for the line through $(x_1,y_1)$ and $(x_2,y_2)$?

Answer: $m=\frac{y_2-y_1}{x_2-x_1}$. Rise over run gives the rate of change between two points.

Flashcard 24: What is the equation of a line in point-slope form through $(x_1,y_1)$ with slope $m$?

Answer: $y-y_1=m(x-x_1)$. Uses a known point and slope to define the line.

Flashcard 25: State the slope-intercept form of a line and identify which value is the $y$-intercept.

Answer: $y=mx+b$; $y$-intercept is $b$. Standard form where $m$ is slope and line crosses $y$-axis at $(0,b)$.

Flashcard 26: What is the $x$-intercept of a graph in terms of the equation $y=f(x)$?

Answer: A solution to $f(x)=0$. The $x$-intercept occurs where the graph crosses the $x$-axis ($y=0$).

Flashcard 27: What is the midpoint formula for endpoints $(x_1,y_1)$ and $(x_2,y_2)$?

Answer: $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$. Average the $x$-coordinates and $y$-coordinates separately.

Flashcard 28: What is the axis of symmetry for $y=a(x-h)^2+k$?

Answer: $x=h$. Vertical line through vertex; parabola symmetric about it.

Flashcard 29: Identify the midpoint of the segment with endpoints $(0,8)$ and $(6,2)$.

Answer: $(3,5)$. Midpoint: $\left(\frac{0+6}{2}, \frac{8+2}{2}\right) = (3,5)$.

Flashcard 30: What is the slope of a line perpendicular to a line with slope $m$ (assume $m\ne 0$)?

Answer: $-\frac{1}{m}$. Perpendicular slopes multiply to $-1$.

Flashcard 31: What is the distance formula between $(x_1,y_1)$ and $(x_2,y_2)$?

Answer: $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Uses Pythagorean theorem on the differences in coordinates.

Flashcard 32: Convert $2x-3y=12$ to slope-intercept form $y=mx+b$.

Answer: $y=\frac{2}{3}x-4$. Isolate $y$: $-3y=-2x+12$, then divide by $-3$.

Flashcard 33: What is the relationship between slopes of perpendicular nonvertical lines?

Answer: $m_1m_2=-1$. Their slopes multiply to negative one.

Flashcard 34: What is the distance between $(0,0)$ and $(3,4)$?

Answer: $5$. Distance formula gives $\sqrt{3^2+4^2}=\sqrt{9+16}=\sqrt{25}=5$.

Flashcard 35: What is the equation of the line through $(4,1)$ and $(4,-3)$?

Answer: $x=4$. Both points have same $x$-coordinate, creating a vertical line.

Flashcard 36: Identify the equation of the line with slope $-2$ passing through $(1,4)$.

Answer: $y=-2x+6$. Using point-slope form: $y - 4 = -2(x - 1)$ simplifies to this.

Flashcard 37: What is the slope of a line perpendicular to a line with slope $m$ (with $m\neq 0$)?

Answer: $-\frac{1}{m}$. Perpendicular slopes multiply to $-1$.

Flashcard 38: What is the slope formula between points $(x_1,y_1)$ and $(x_2,y_2)$?

Answer: $m=\frac{y_2-y_1}{x_2-x_1}$. Rise over run: change in $y$ divided by change in $x$.

Flashcard 39: What is the slope of a vertical line (for example, $x=-2$)?

Answer: Undefined. Vertical lines have no horizontal change, making slope undefined.

Flashcard 40: Identify the $x$-intercept of the line $2x+3y=12$.

Answer: $(6,0)$. Set $y=0$: $2x+0=12$, so $x=6$.

Flashcard 41: Identify the $y$-intercept of the line $3x+2y=12$.

Answer: $(0,6)$. Set $x=0$ and solve: $2y=12$, so $y=6$.

Flashcard 42: What is the equation of the line with slope $3$ passing through $(0,-2)$?

Answer: $y=3x-2$. Substitute slope and point $(0,-2)$ into $y=mx+b$.

Flashcard 43: What is the midpoint formula for the segment joining $(x_1,y_1)$ and $(x_2,y_2)$?

Answer: $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$. Average the $x$-coordinates and $y$-coordinates separately.

Flashcard 44: What is the slope-intercept form of a line with slope $m$ and $y$-intercept $b$?

Answer: $y=mx+b$. $m$ is slope, $b$ is where line crosses $y$-axis.

Flashcard 45: What are the coordinates of the origin on a coordinate plane?

Answer: $(0,0)$. The origin is where both axes intersect at zero.

Flashcard 46: Identify the $y$-intercept of the line $2x+3y=12$.

Answer: $(0,4)$. Set $x=0$: $0+3y=12$, so $y=4$.

Flashcard 47: Find the slope of the line through $(2,1)$ and $(6,9)$.

Answer: $2$. $m=\frac{9-1}{6-2}=\frac{8}{4}=2$.

Flashcard 48: What is the slope of a line perpendicular to $y=4x-5$?

Answer: $m=-\frac{1}{4}$. Perpendicular slopes multiply to $-1$: $4 \times (-\frac{1}{4})=-1$.

Flashcard 49: Identify the $x$-intercept of the line $4x-5y=20$.

Answer: $(5,0)$. Set y = 0: $4x = 20$, so $x = 5$.

Flashcard 50: What are the coordinates of the origin on the coordinate plane?

Answer: $(0,0)$. The origin is where the x-axis and y-axis intersect.

Flashcard 51: Identify the distance between $(0,0)$ and $(3,4)$.

Answer: $5$. $\sqrt{(3-0)^2+(4-0)^2}=\sqrt{9+16}=\sqrt{25}=5$.

Flashcard 52: What is the distance formula between points $(x_1,y_1)$ and $(x_2,y_2)$?

Answer: $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Uses Pythagorean theorem on the differences in coordinates.

Flashcard 53: What is the slope relationship of perpendicular nonvertical lines?

Answer: $m_1m_2=-1$. Product of perpendicular slopes always equals negative one.

Flashcard 54: What is the equation of a horizontal line passing through $y=b$?

Answer: $y=b$. Horizontal lines have constant $y$-value for all $x$.

Flashcard 55: What is the midpoint formula for points $(x_1,y_1)$ and $(x_2,y_2)$?

Answer: $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$. Average the x-coordinates and y-coordinates.

Flashcard 56: What is the $y$-intercept of a line written in slope-intercept form $y=mx+b$?

Answer: $(0,b)$. The y-intercept occurs where the line crosses the y-axis (when $x=0$).

Flashcard 57: Identify the slope of the line $3x+2y=8$.

Answer: $m=-\frac{3}{2}$. Rewrite as $y=-\frac{3}{2}x+4$ to identify slope.

Flashcard 58: What is the $y$-intercept of a line written as $y=mx+b$?

Answer: $(0,b)$. The y-intercept occurs when x = 0, leaving y = b.

Flashcard 59: What is the slope of a line perpendicular to a line with slope $\frac{3}{4}$?

Answer: $-\frac{4}{3}$. Perpendicular slopes multiply to $-1$: $\frac{3}{4}\cdot(-\frac{4}{3})=-1$.

Flashcard 60: What is the distance formula between $(x_1,y_1)$ and $(x_2,y_2)$?

Answer: $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Apply the Pythagorean theorem to find straight-line distance.

Flashcard 61: What is the $x$-intercept of the line $2x+3y=6$?

Answer: $(3,0)$. Set $y=0$ and solve: $2x+3(0)=6$, so $x=3$.

Flashcard 62: What is the slope of a horizontal line given by $y=c$?

Answer: $0$. Horizontal lines have no vertical change, so slope is zero.

Flashcard 63: What is the slope of a horizontal line (in simplest form)?

Answer: $0$. No vertical change means rise = 0, so slope = 0.

Flashcard 64: What is the slope of a line perpendicular to a line with slope $\frac{2}{3}$?

Answer: $-\frac{3}{2}$. Take negative reciprocal: $-\frac{1}{\frac{2}{3}}=-\frac{3}{2}$.

Flashcard 65: Identify the vertex of $y=(x-4)^2-9$.

Answer: $(4,-9)$. In vertex form, $(h,k)$ gives the vertex directly.

Flashcard 66: What is the $x$-coordinate of the vertex of $y=2x^2-8x+1$?

Answer: $2$. Use $x=-\frac{b}{2a}$ with $a=2$, $b=-8$: $x=\frac{8}{4}=2$.

Flashcard 67: What is the slope of a horizontal line of the form $y=c$?

Answer: $0$. Horizontal lines have no vertical change, so slope is zero.

Flashcard 68: What is the equation of the line with slope $3$ that passes through $(0,-2)$?

Answer: $y=3x-2$. Point $(0,-2)$ gives $y$-intercept $b=-2$ in $y=mx+b$.

Flashcard 69: A line has slope $\frac{2}{3}$. What is the slope of a perpendicular line?

Answer: $-\frac{3}{2}$. Negative reciprocal: $-1÷\frac{2}{3}=-\frac{3}{2}$.

Flashcard 70: What is the slope of a vertical line written as $x=c$?

Answer: Undefined. Vertical lines have no run, making slope undefined.

Flashcard 71: What is the average rate of change of $f$ from $x=a$ to $x=b$?

Answer: $\frac{f(b)-f(a)}{b-a}$. Slope formula applied to function values at two points.

Flashcard 72: What is the slope-intercept form of a line?

Answer: $y=mx+b$. $m$ is slope and $b$ is the $y$-intercept.

Flashcard 73: What is the slope formula for points $(x_1,y_1)$ and $(x_2,y_2)$?

Answer: $m=\frac{y_2-y_1}{x_2-x_1}$. Rise over run: change in $y$ divided by change in $x$.

Flashcard 74: Identify the slope of a line perpendicular to a line with slope $\frac{2}{3}$.

Answer: $-\frac{3}{2}$. Perpendicular slopes multiply to $-1$: $\frac{2}{3} \cdot (-\frac{3}{2})=-1$.

Flashcard 75: What is the slope of the line $4x-2y=10$?

Answer: $2$. Rewrite as $y=2x-5$, so slope is the coefficient of $x$.

Flashcard 76: Identify the $y$-intercept of the line $4x-5y=10$ as an ordered pair.

Answer: $(0,-2)$. Set $x=0$: $-5y=10$, so $y=-2$.

Flashcard 77: What is the $x$-intercept of the line $y=mx+b$ (as a formula, assuming $m\ne 0$)?

Answer: $x=-\frac{b}{m}$. Set $y=0$ and solve for $x$ to find where line crosses $x$-axis.

Flashcard 78: What is the slope of the line passing through $(2,3)$ and $(6,11)$?

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

Flashcard 79: Identify the vertex of the parabola $y=(x-2)^2+5$.

Answer: $(2,5)$. Vertex form $(x-h)^2+k$ has vertex at $(h,k)$.

Flashcard 80: What is the slope of a vertical line given by $x=-3$?

Answer: Undefined. Vertical lines have infinite steepness.

Flashcard 81: What is the $x$-intercept of the line $y=-2x+6$?

Answer: $(3,0)$. Set $y=0$: $0=-2x+6$, so $x=3$.

Flashcard 82: Find the $y$-intercept $b$ of the line $y=3x-7$.

Answer: $-7$. In $y=mx+b$, the constant term $b$ is the y-intercept.

Flashcard 83: Identify the equation of the line through $(1,-2)$ with slope $5$.

Answer: $y=5x-7$. Use point-slope: $y-(-2)=5(x-1)$, then simplify.

Flashcard 84: Identify the equation of the line through $(4,-1)$ with slope $2$.

Answer: $y=2x-9$. Using point-slope form: $y-(-1)=2(x-4)$ simplifies to $y=2x-9$.

Flashcard 85: What is the standard form of a line (with integer coefficients)?

Answer: $Ax+By=C$. $A$, $B$, and $C$ are integers with no common factors.

Flashcard 86: Identify the slope and $y$-intercept of $y=-2x+5$.

Answer: $m=-2,\ b=5$. Read directly from slope-intercept form.

Flashcard 87: What is the slope of a vertical line given by $x=c$?

Answer: Undefined. Vertical lines have zero run, making slope undefined.

Flashcard 88: Identify the $x$-intercept of $y=mx+b$ in terms of $m$ and $b$ (assume $m\ne 0$).

Answer: $\left(-\frac{b}{m},0\right)$. Set $y=0$ and solve for $x$ to find where line crosses $x$-axis.

Flashcard 89: Identify the transformation of $y=f(-x)$ relative to $y=f(x)$.

Answer: Reflect across the $y$-axis. Negative input flips graph across vertical axis.

Flashcard 90: What is the slope of the line passing through $(2,5)$ and $(6,1)$?

Answer: $-1$. Using slope formula: $\frac{1-5}{6-2}=\frac{-4}{4}=-1$.

Flashcard 91: What is the $y$-intercept of a line written as $ax+by=c$ (assume $b\neq 0$)?

Answer: $\left(0,\frac{c}{b}\right)$. Set x = 0 and solve for y to find where line crosses y-axis.

Flashcard 92: Identify the vertex of the parabola $y=(x-3)^2+4$.

Answer: $(3,4)$. Vertex form $y=(x-h)^2+k$ has vertex at $(h,k)$.

Flashcard 93: Identify the slope of the line passing through $(2,5)$ and $(6,1)$.

Answer: $m=-1$. Using slope formula: $\frac{1-5}{6-2}=\frac{-4}{4}=-1$.

Flashcard 94: What are the coordinates of the origin on a coordinate plane?

Answer: $(0,0)$. The origin is where the x-axis and y-axis intersect.

Flashcard 95: Identify the $y$-intercept of $y=-\frac{1}{2}x+6$.

Answer: $(0,6)$. When $x=0$, $y=-\frac{1}{2}(0)+6=6$.

Flashcard 96: What is the midpoint of the segment with endpoints $(-2,5)$ and $(6,1)$?

Answer: $(2,3)$. Apply midpoint formula: $\left(\frac{-2+6}{2},\frac{5+1}{2}\right)=(2,3)$.

Flashcard 97: Identify the transformation of $y=f(x-h)$ relative to $y=f(x)$.

Answer: Shift right $h$ units. Subtracting $h$ from input shifts graph horizontally right.

Flashcard 98: State the formula for the midpoint of a segment with endpoints $(x_1,y_1)$ and $(x_2,y_2)$.

Answer: $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$. Average the x-coordinates and y-coordinates separately.

Flashcard 99: What is the slope of any vertical line (for example, $x=-3$)?

Answer: Slope is undefined. Vertical lines have no run, making division impossible.

Flashcard 100: What is the slope of a horizontal line given by $y=7$?

Answer: $0$. Horizontal lines have no vertical change.