ACT Math Flashcards: Linear Functions

What this deck covers

This deck focuses on Linear Functions, giving you a quick way to review the definitions, rules, and examples that matter most for ACT 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 point-slope form of a linear equation through $(x_1,y_1)$ with slope $m$?

Answer: $y-y_1=m(x-x_1)$. Uses known point $(x_1,y_1)$ and slope $m$.

Flashcard 2: What is the equation of the line with $x$-intercept $-2$ and $y$-intercept $6$?

Answer: $y=3x+6$. Slope is $\frac{6-0}{0-(-2)}=3$.

Flashcard 3: What is the y-intercept of the equation $4x + 2y = 8$?

Answer:

1. Set $x = 0$: $2y = 8$, so $y = 4$.

Flashcard 4: Is the line $y = \frac{3}{4}x - 2$ increasing or decreasing?

Answer: Increasing. Positive slope means line goes up left to right.

Flashcard 5: What is the $y$-intercept of the line $y=mx+b$?

Answer: $b$. The constant term when $x=0$.

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

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

Flashcard 7: Find the x-intercept of $4x - y = 8$.

Answer:

1. Set $y = 0$: $4x = 8$, so $x = 2$.

Flashcard 8: What is the equation of the line through points $(2,1)$ and $(2,5)$?

Answer: $x=2$. Same $x$-coordinate means vertical line.

Flashcard 9: Find the y-intercept of the line $5x + 2y = 10$.

Answer:

1. Set $x = 0$: $2y = 10$, so $y = 5$.

Flashcard 10: What is the slope-intercept form of a linear equation?

Answer: $y = mx + b$. Form showing slope $m$ and y-intercept $b$ directly.

Flashcard 11: What is the average rate of change of $f(x)=-3x+2$ from $x=1$ to $x=4$?

Answer: $-3$. For linear functions, rate of change equals slope.

Flashcard 12: What is the $y$-intercept of the line through $(2,3)$ with slope $4$?

Answer: $-5$. Use point-slope: $3=4(2)+b$, so $b=-5$.

Flashcard 13: What is the equation of the line through points $(1,2)$ and $(3,6)$ in slope-intercept form?

Answer: $y=2x$. Slope is $2$, passes through origin.

Flashcard 14: Is the line $y = \frac{3}{4}x - 2$ increasing or decreasing?

Answer: Increasing. Positive slope means line goes up left to right.

Flashcard 15: What is the $x$-intercept of $Ax+By=C$ (with $A\ne 0$)?

Answer: $\frac{C}{A}$. Set $y=0$ in standard form and solve.

Flashcard 16: What is the average rate of change of $f(x)=-3x+2$ from $x=1$ to $x=4$?

Answer: $-3$. For linear functions, rate of change equals slope.

Flashcard 17: Determine if the lines $y = 2x + 3$ and $y = 2x - 4$ are parallel.

Answer: Yes. Both have slope $2$, so they're parallel.

Flashcard 18: Determine if the lines $2x + 3y = 6$ and $4x + 6y = 12$ are identical.

Answer: Yes. Second equation is twice the first; same line.

Flashcard 19: What is $f(5)$ if $f(x)=2x-3$?

Answer: $7$. Substitute: $f(5)=2(5)-3=7$.

Flashcard 20: Determine if $y = \frac{3}{2}x + 5$ and $y = -\frac{2}{3}x + 1$ are perpendicular.

Answer: Yes. Slopes $\frac{3}{2}$ and $-\frac{2}{3}$ multiply to $-1$.

Flashcard 21: What is the $x$-intercept of $Ax+By=C$ (with $A\ne 0$)?

Answer: $\frac{C}{A}$. Set $y=0$ in standard form and solve.

Flashcard 22: What is the slope of a line through $(2, 3)$ and $(4, 3)$?

Answer:

1. Same $y$-coordinate means horizontal line.

Flashcard 23: What is the $y$-intercept of $Ax+By=C$ (with $B \neq 0$)?

Answer: $\frac{C}{B}$. Set $x=0$ in standard form and solve.

Flashcard 24: What is the slope of the line segment from $(4,-2)$ to $(0,6)$?

Answer: $-2$. Using slope formula: $\frac{6-(-2)}{0-4}=\frac{8}{-4}=-2$.

Flashcard 25: What is the equation of the line passing through $(-3,2)$ with undefined slope?

Answer: $x=-3$. Undefined slope means vertical line.

Flashcard 26: What is the $y$-intercept of the line $4x-2y=10$?

Answer: $-5$. Convert to slope-intercept: $y=2x-5$.

Flashcard 27: What is the $x$-intercept of the line $y=mx+b$ in terms of $m$ and $b$ (assuming $m\ne 0$)?

Answer: $x=-\frac{b}{m}$. Set $y=0$ and solve for $x$.

Flashcard 28: What is the equation of the line through $(2,3)$ with slope $4$ in slope-intercept form?

Answer: $y=4x-5$. From point-slope: $y-3=4(x-2)$.

Flashcard 29: What is the equation of a line passing through $(0, -2)$ with slope $-3$?

Answer: $y = -3x - 2$. Line passes through $(0, -2)$ with given slope.

Flashcard 30: Convert $y = 2x + 3$ to standard form.

Answer: $-2x + y = 3$. Move $x$ term to left side.

Flashcard 31: What is the slope-intercept form of a linear equation?

Answer: $y = mx + b$. Form showing slope $m$ and y-intercept $b$ directly.

Flashcard 32: What is the equation of the line passing through $(1,5)$ with slope $0$?

Answer: $y=5$. Zero slope means horizontal line.

Flashcard 33: Convert the equation $y = -2x + 5$ to standard form.

Answer: $2x + y = 5$. Move $x$ term: $2x + y = 5$.

Flashcard 34: What is the slope of the line represented by $Ax+By=C$ (with $B\ne 0$)?

Answer: $m=-\frac{A}{B}$. Rearrange to slope-intercept form.

Flashcard 35: What is the slope of a horizontal line?

Answer: $0$. No vertical change means zero slope.

Flashcard 36: Find the slope of the line passing through $(0, 0)$ and $(5, 10)$.

Answer:

1. Using slope formula: $\frac{10-0}{5-0} = 2$.

Flashcard 37: What is the equation of a horizontal line passing through $y=k$?

Answer: $y=k$. Constant $y$-value for all $x$.

Flashcard 38: What is the $y$-intercept of the line $y=\frac{1}{4}x-8$?

Answer: $-8$. The constant term in slope-intercept form.

Flashcard 39: What is the equation of the line through points $(-3,4)$ and $(1,4)$?

Answer: $y=4$. Same $y$-coordinate means horizontal line.

Flashcard 40: What is the equation of the line passing through $(-3,2)$ with undefined slope?

Answer: $x=-3$. Undefined slope means vertical line.

Flashcard 41: What is the standard form of a linear equation?

Answer: $Ax + By = C$. General form with integer coefficients $A$, $B$, and $C$.

Flashcard 42: What is the slope of the line passing through $(2,5)$ and $(6,13)$?

Answer: $2$. Using slope formula: $\frac{13-5}{6-2}=\frac{8}{4}=2$.

Flashcard 43: What is the slope of a vertical line?

Answer: Undefined. Division by zero in slope formula.

Flashcard 44: Find the x-intercept of $4x - y = 8$.

Answer:

1. Set $y = 0$: $4x = 8$, so $x = 2$.

Flashcard 45: Find the slope between points $(2, 3)$ and $(4, 7)$.

Answer:

1. Using slope formula: $\frac{7-3}{4-2} = \frac{4}{2} = 2$.

Flashcard 46: What condition on slopes $m_1$ and $m_2$ means two nonvertical lines are perpendicular?

Answer: $m_1m_2=-1$. Perpendicular slopes are negative reciprocals.

Flashcard 47: Determine if $y = \frac{3}{2}x + 5$ and $y = -\frac{2}{3}x + 1$ are perpendicular.

Answer: Yes. Slopes $\frac{3}{2}$ and $-\frac{2}{3}$ multiply to $-1$.

Flashcard 48: What is the slope-intercept form of the line $2x+3y=12$?

Answer: $y=-\frac{2}{3}x+4$. Divide by $3$: $y=-\frac{2}{3}x+4$.

Flashcard 49: What is the slope of a line parallel to the line $y = \frac{5}{2}x + 1$?

Answer: $\frac{5}{2}$. Parallel lines have identical slopes.

Flashcard 50: State the formula to calculate the slope between two points $(x_1, y_1)$ and $(x_2, y_2)$.

Answer: $m = \frac{y_2 - y_1}{x_2 - x_1}$. Rise over run between any two points.

Flashcard 51: What is the $y$-intercept of the line $y=mx+b$?

Answer: $b$. The constant term when $x=0$.

Flashcard 52: What is $x$ if the point $(x,9)$ lies on the line $y=2x+1$?

Answer: $4$. Solve: $9=2x+1$, so $x=4$.

Flashcard 53: Find the x-intercept of the line $3x - y = 9$.

Answer:

1. Set $y = 0$: $3x = 9$, so $x = 3$.

Flashcard 54: Find the slope of the line $x = 5$.

Answer: Undefined. Vertical line has undefined slope.

Flashcard 55: What is the x-intercept of the equation $y = 3x + 9$?

Answer: -3. Set $y = 0$: $0 = 3x + 9$, so $x = -3$.

Flashcard 56: What is the x-intercept of the equation $y = 3x + 9$?

Answer: -3. Set $y = 0$: $0 = 3x + 9$, so $x = -3$.

Flashcard 57: Determine if the lines $y = 2x + 3$ and $y = 2x - 4$ are parallel.

Answer: Yes. Both have slope $2$, so they're parallel.

Flashcard 58: Convert $3x + 4y = 12$ to slope-intercept form.

Answer: $y = -\frac{3}{4}x + 3$. Solve for $y$: $4y = -3x + 12$.

Flashcard 59: What is the slope of a line parallel to the line $y = \frac{5}{2}x + 1$?

Answer: $\frac{5}{2}$. Parallel lines have identical slopes.

Flashcard 60: Convert $3x + 4y = 12$ to slope-intercept form.

Answer: $y = -\frac{3}{4}x + 3$. Solve for $y$: $4y = -3x + 12$.

Flashcard 61: Find the slope of a horizontal line.

Answer:

1. No vertical change means slope equals zero.

Flashcard 62: What is the $y$-intercept of the line through $(2,3)$ with slope $4$?

Answer: $-5$. Use point-slope: $3=4(2)+b$, so $b=-5$.

Flashcard 63: What is the equation of the line with slope $-4$ passing through $(0,7)$?

Answer: $y=-4x+7$. Point $(0,7)$ gives $y$-intercept directly.

Flashcard 64: Find the y-intercept of the line $5x + 2y = 10$.

Answer:

1. Set $x = 0$: $2y = 10$, so $y = 5$.

Flashcard 65: What is the slope of the line passing through $(2,5)$ and $(6,13)$?

Answer: $2$. Using slope formula: $\frac{13-5}{6-2}=\frac{8}{4}=2$.

Flashcard 66: What is the slope-intercept form of a linear equation?

Answer: $y=mx+b$. Standard form where $m$ is slope and $b$ is $y$-intercept.

Flashcard 67: What is the slope of the line $y=-7$?

Answer: $0$. Horizontal line has zero slope.

Flashcard 68: Identify the slope of the line $3y - 9x = 12$.

Answer:

1. Rewrite as $y = 3x + 4$; slope is $3$.

Flashcard 69: What condition on slopes $m_1$ and $m_2$ means two nonvertical lines are perpendicular?

Answer: $m_1m_2=-1$. Perpendicular slopes are negative reciprocals.

Flashcard 70: What is the equation in slope-intercept form of the line with slope $3$ and $y$-intercept $-2$?

Answer: $y=3x-2$. Direct substitution into $y=mx+b$.

Flashcard 71: What is the slope of the line through $(0,-1)$ and $(8,3)$?

Answer: $\frac{1}{2}$. Using slope formula: $\frac{3-(-1)}{8-0}=\frac{4}{8}=\frac{1}{2}$.

Flashcard 72: What is the average rate of change of a linear function $f$ on $[a,b]$?

Answer: $\frac{f(b)-f(a)}{b-a}$. For linear functions, this equals the slope.

Flashcard 73: What is the slope-intercept form of the line $2x+3y=12$?

Answer: $y=-\frac{2}{3}x+4$. Divide by $3$: $y=-\frac{2}{3}x+4$.

Flashcard 74: What is the slope of the line given by the equation $4x - 2y = 8$?

Answer:

1. Rewrite as $y = 2x - 4$; slope is $2$.

Flashcard 75: What is the slope of the line $x=9$?

Answer: Undefined. Vertical line has undefined slope.

Flashcard 76: Find the slope of a vertical line.

Answer: Undefined. Division by zero in slope formula.

Flashcard 77: Convert the equation $2x + 3y = 6$ to slope-intercept form.

Answer: $y = -\frac{2}{3}x + 2$. Solve for $y$: $3y = -2x + 6$, so $y = -\frac{2}{3}x + 2$.

Flashcard 78: What is the slope of the line passing through $(-1,4)$ and $(3,0)$?

Answer: $-1$. Using slope formula: $\frac{0-4}{3-(-1)}=\frac{-4}{4}=-1$.

Flashcard 79: What is the equation of the line through $(2, -1)$ with slope $\frac{1}{2}$?

Answer: $y + 1 = \frac{1}{2}(x - 2)$. Point-slope form with given point and slope.

Flashcard 80: What is the slope of the line passing through $(-1,4)$ and $(3,0)$?

Answer: $-1$. Using slope formula: $\frac{0-4}{3-(-1)}=\frac{-4}{4}=-1$.

Flashcard 81: What is the $y$-intercept of the line $y=\frac{1}{4}x-8$?

Answer: $-8$. The constant term in slope-intercept form.

Flashcard 82: What is the equation of the line with slope $2$ passing through $(3,-1)$ in point-slope form?

Answer: $y+1=2(x-3)$. Direct substitution into point-slope form.

Flashcard 83: Determine if $y = 4x + 1$ and $y = 4x - 3$ are parallel.

Answer: Yes. Both have slope $4$, so they're parallel.

Flashcard 84: What is the equation of a line with slope 2 passing through $(1, 3)$?

Answer: $y - 3 = 2(x - 1)$. Point-slope form: $y - y_1 = m(x - x_1)$.

Flashcard 85: What is the slope of the line $y = 7$?

Answer:

1. Horizontal line has zero slope.

Flashcard 86: What is the equation of a line passing through $(0, -2)$ with slope $-3$?

Answer: $y = -3x - 2$. Line passes through $(0, -2)$ with given slope.

Flashcard 87: What is the point-slope form of a linear equation through $(x_1,y_1)$ with slope $m$?

Answer: $y-y_1=m(x-x_1)$. Uses known point $(x_1,y_1)$ and slope $m$.

Flashcard 88: Convert the equation $2x + 3y = 6$ to slope-intercept form.

Answer: $y = -\frac{2}{3}x + 2$. Solve for $y$: $3y = -2x + 6$, so $y = -\frac{2}{3}x + 2$.

Flashcard 89: Is the line $y = -\frac{1}{2}x + 3$ increasing or decreasing?

Answer: Decreasing. Negative slope means line goes down left to right.

Flashcard 90: What is the slope of a horizontal line?

Answer: $0$. No vertical change means zero slope.

Flashcard 91: What is the $y$-intercept of $Ax+By=C$ (with $B \neq 0$)?

Answer: $\frac{C}{B}$. Set $x=0$ in standard form and solve.

Flashcard 92: What is the equation of the line with slope $2$ passing through $(3,-1)$ in point-slope form?

Answer: $y+1=2(x-3)$. Direct substitution into point-slope form.

Flashcard 93: What condition on slopes $m_1$ and $m_2$ means two nonvertical lines are parallel?

Answer: $m_1=m_2$. Parallel lines have identical slopes.

Flashcard 94: Identify the y-intercept in the equation $y = 4x - 7$.

Answer: -7. Constant term in slope-intercept form.

Flashcard 95: What is the standard form of a linear equation?

Answer: $Ax + By = C$. General form with integer coefficients $A$, $B$, and $C$.

Flashcard 96: What is the $y$-intercept of the line $4x-2y=10$?

Answer: $-5$. Convert to slope-intercept: $y=2x-5$.

Flashcard 97: What does the variable $b$ represent in the equation $y = mx + b$?

Answer: The y-intercept of the line. Constant term where line crosses y-axis at $(0, b)$.

Flashcard 98: What is the equation of a vertical line passing through $x=h$?

Answer: $x=h$. Constant $x$-value for all $y$.

Flashcard 99: What is the equation of the line with slope $-1$ and $x$-intercept $4$?

Answer: $y=-x+4$. Line passes through $(4,0)$ and $(0,4)$.

Flashcard 100: What is the equation of a horizontal line passing through $y=k$?

Answer: $y=k$. Constant $y$-value for all $x$.