ISEE Upper Level Mathematics Achievement Flashcards: Finding Slope

What this deck covers

This deck focuses on Finding Slope, giving you a quick way to review the definitions, rules, and examples that matter most for ISEE Upper Level Mathematics Achievement.

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 slope of the line through $(-6,1)$ and $(0,-2)$?

Answer: $m=-\frac{1}{2}$. Compute $(-2-1)/(0-(-6))=-3/6=-1/2$.

Flashcard 2: What happens to the slope value if you swap the two points in $m=\frac{y_2-y_1}{x_2-x_1}$?

Answer: Slope stays the same. Swapping points negates both numerator and denominator, preserving the slope value.

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

Answer: $m=-2$. Slope is $(0-8)/(1-(-3))=-8/4=-2$.

Flashcard 4: What is the slope of the line through $(2,5)$ and $(-1,-4)$?

Answer: $m=3$. Using $(-4-5)/(-1-2)=-9/(-3)=3$.

Flashcard 5: What is the slope of the line through $(4,7)$ and $(0,1)$?

Answer: $m=\frac{3}{2}$. Calculate $(1-7)/(0-4)=-6/(-4)=3/2$.

Flashcard 6: State the slope formula for the line through points $(x_1,y_1)$ and $(x_2,y_2)$.

Answer: $m=\frac{y_2-y_1}{x_2-x_1}$. The slope measures the rate of change as the vertical difference divided by the horizontal difference between two points.

Flashcard 7: What is the slope of the line through $(-5,2)$ and $(-1,10)$?

Answer: $m=2$. Formula gives $(10-2)/(-1-(-5))=8/4=2$.

Flashcard 8: What is the slope of the line through $(0,0)$ and $(5,-10)$?

Answer: $m=-2$. Calculate $(-10-0)/(5-0)=-10/5=-2$, showing a downward slope.

Flashcard 9: What is the slope of the line through $(-2,9)$ and $(-8,3)$?

Answer: $m=1$. Compute $(3-9)/(-8-(-2))=-6/(-6)=1$.

Flashcard 10: What is the slope of the line through $(5,12)$ and $(1,4)$?

Answer: $m=2$. Order of points yields $(4-12)/(1-5)=-8/(-4)=2$.

Flashcard 11: What is the slope of the line through $(-4,-6)$ and $(2,3)$?

Answer: $m=\frac{3}{2}$. Using $(3-(-6))/(2-(-4))=9/6=3/2$.

Flashcard 12: What is the slope of the line through $(-1,4)$ and $(3,-2)$?

Answer: $m=-\frac{3}{2}$. Using the formula, $(-2-4)/(3-(-1))=-6/4=-3/2$, indicating a negative slope.

Flashcard 13: What is the slope of the line through $(1,-2)$ and $(4,4)$?

Answer: $m=2$. Compute $(4-(-2))/(4-1)=6/3=2$ using the slope formula.

Flashcard 14: What is the slope of the line through $(2,3)$ and $(6,11)$?

Answer: $m=2$. Apply the slope formula: $(11-3)/(6-2)=8/4=2$, representing rise over run.

Flashcard 15: Find and correct the error: $m=\frac{x_2-x_1}{y_2-y_1}$ for slope between two points.

Answer: Correct: $m=\frac{y_2-y_1}{x_2-x_1}$. The incorrect formula inverts rise over run; the correct one uses change in y over change in x.

Flashcard 16: What is the slope of the line through $(8,-3)$ and $(2,9)$?

Answer: $m=-2$. Calculate $(9-(-3))/(2-8)=12/(-6)=-2$, a negative value.

Flashcard 17: What is the slope of the line through $(-3,5)$ and $(4,5)$?

Answer: $m=0$. Identical y-coordinates yield a numerator of zero, resulting in a slope of zero for horizontal lines.

Flashcard 18: Identify the slope type for a vertical line (in terms of $m$).

Answer: Slope is undefined. Vertical lines have no change in x, causing division by zero and an undefined slope.

Flashcard 19: What is the slope of the line through $(-2,-1)$ and $(2,7)$?

Answer: $m=2$. Using $(7-(-1))/(2-(-2))=8/4=2$, the slope is positive.

Flashcard 20: What is the slope of the line through $(7,2)$ and $(7,9)$?

Answer: Slope is undefined. Identical x-coordinates result in division by zero, making the slope undefined for vertical lines.

Flashcard 21: What is the slope of the line through $(3,1)$ and $(9,0)$?

Answer: $m=-\frac{1}{6}$. Slope is $(0-1)/(9-3)=-1/6$, showing a slight decline.

Flashcard 22: Which condition makes the slope undefined when using $m=\frac{y_2-y_1}{x_2-x_1}$?

Answer: $x_2-x_1=0$. A zero denominator indicates a vertical line, where slope cannot be defined.

Flashcard 23: What is the slope of the line through $(2,-5)$ and $(10,-1)$?

Answer: $m=\frac{1}{2}$. Formula results in $(-1-(-5))/(10-2)=4/8=1/2$.

Flashcard 24: Which condition makes the slope equal to $0$ when using $m=\frac{y_2-y_1}{x_2-x_1}$?

Answer: $y_2-y_1=0$. A zero numerator indicates no vertical change, yielding a horizontal line with zero slope.

Flashcard 25: Identify the slope type for a horizontal line (in terms of $m$).

Answer: $m=0$. Horizontal lines have no change in y, so the slope is zero regardless of x-change.