Analyze and Sketch Function Graphs - 8th Grade Math

1. Which classification fits the graph of $y=3x-2$: linear or nonlinear?

   Linear. First-degree equations produce straight lines.

2. What does a negative slope mean about the relationship between $x$ and $y$?

   Negative relationship: as $x$ increases, $y$ decreases. Variables move in opposite directions.

3. What qualitative graph matches "increases at a constant rate"?

   A straight line rising left to right (linear with $m>0$). Constant positive slope creates steady upward line.

4. Which interval shows the function decreasing for points $(1,2)$, $(3,5)$, $(6,4)$?

   Decreasing on $3\le x\le 6$. $y$ goes from 5 to 4 as $x$ goes from 3 to 6.

5. Which sketch matches: decreasing at a constant rate and crossing the $x$-axis at $(3,0)$?

   A downward straight line passing through $(3,0)$. Constant negative slope hitting $x$-axis at $3$.

6. What does a constant rate of change look like on a graph?

   A straight line with the same slope everywhere. Constant rate means uniform steepness throughout.

7. Identify whether the function is increasing, decreasing, or constant from $(0,3)$ to $(4,1)$.

   Decreasing (the $y$-values go from $3$ down to $1$). Compare $y$-values: $3$ to $1$ is a decrease.

8. What is the $y$-intercept on a graph, stated using coordinates?

   The point where $x=0$, written as $(0,b)$. Where the graph crosses the $y$-axis.

9. What does a positive slope mean about the relationship between $x$ and $y$?

   Positive relationship: as $x$ increases, $y$ increases. Both variables move in the same direction.

10. What qualitative change occurs if a graph switches from rising to falling at one point?

    It changes from increasing to decreasing at a turning point (maximum). A peak where direction reverses.

11. Identify the $x$-values where the function is constant: points $(0,2)$, $(3,2)$, $(5,4)$ are connected in order.

    Constant on $0<x<3$. $y$ stays at $2$ from $x=0$ to $x=3$.

12. What is the qualitative meaning of a steeper line compared to a less steep line?

    The rate of change has a greater magnitude (larger $|m|$). Steeper lines change faster per unit of $x$.

13. Which statement best defines a decreasing function on an interval of $x$?

    As $x$ increases, $y$ decreases on that interval. The variables move in opposite directions.

14. Which classification fits the graph of $y=x^2$: linear or nonlinear?

    Nonlinear. Quadratic functions produce parabolas, which are curved.

15. What is the qualitative meaning of a positive slope on a graph of $y$ versus $x$?

    $y$ increases as $x$ increases. Positive slope means upward trend from left to right.

16. What is the $x$-intercept on a graph, stated using coordinates?

    A point where $y=0$, written as $(a,0)$. Where the graph crosses the $x$-axis.

17. What does it mean for a function to be constant on an interval in a graph?

    As $x$ changes, $y$ stays the same (a horizontal segment). Horizontal line has zero slope, so $y$ doesn't change.

18. Identify the $x$-values where the function is increasing: points $(0,1)$, $(2,3)$, $(4,2)$, $(6,5)$ are connected in order.

    Increasing on $0<x<2$ and $4<x<6$. $y$ goes from $1$ to $3$ then $2$ to $5$ in those intervals.

19. What is the slope between points $(2,1)$ and $(6,9)$?

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

20. What is the $y$-intercept on a graph, in terms of $x$ and $y$?

    The point where $x=0$; it is $(0,b)$. The $y$-intercept occurs where the graph crosses the $y$-axis.

21. What does a horizontal line on a graph mean about the function's rate of change?

    Zero rate of change: slope $m=0$ (constant function). Horizontal lines have no vertical change.

22. Identify the slope from points $(1,2)$ and $(5,10)$ using $m=\frac{\Delta y}{\Delta x}$.

    $m=\frac{10-2}{5-1}=2$. Apply slope formula: rise over run.

23. What is the slope of the line through $(0,1)$ and $(2,5)$?

    $m=\frac{5-1}{2-0}=2$. Use slope formula: $m=\frac{y_2-y_1}{x_2-x_1}$.

24. What feature on a graph shows a nonlinear function?

    It is not a straight line; the slope (rate of change) changes. Curves or bends indicate changing rates of change.

25. Which sketch description matches: "Starts low, increases slowly, then rises faster and faster"?

    An increasing curve with slope increasing as $x$ increases. Shows exponential growth behavior.

26. What feature on a graph shows a linear function (not just an interval)?

    A straight line with a constant rate of change (constant slope). Linear functions have constant slope throughout.

27. Identify the slope sign when a graph falls from left to right.

    Negative slope: $m<0$. Falling lines have negative slopes.

28. What graph feature shows a function is nonlinear rather than linear?

    A nonlinear function graphs as a curve (not a straight line). Nonlinear functions have varying rates of change.

29. What qualitative graph matches "starts high, decreases quickly, then levels off"?

    A decreasing curve that flattens (slope negative, $|m|$ shrinking). Rapid decrease followed by gradual leveling.

30. Identify whether the function is increasing, decreasing, or constant if its line has slope $m=-4$.

    Decreasing. Negative slope means the line goes down from left to right.

31. What does it mean if a graph becomes less steep (flattens) as $x$ increases?

    The rate of change is decreasing; $y$ changes slower per $1$ in $x$. Flatter slope means slower rate of change.

32. What is the slope interpretation when a line rises from left to right?

    Positive slope: $m>0$. Rising lines have positive slopes by definition.

33. What does it mean for a function to be increasing on an interval in a graph?

    As $x$ increases, $y$ increases (the graph rises left to right). Rising graph means positive change in $y$ as $x$ grows.

34. Identify the slope sign when a graph rises from left to right.

    Positive slope: $m>0$. Rising lines have positive slopes.

35. Which statement best defines an increasing function on an interval of $x$?

    As $x$ increases, $y$ increases on that interval. Both variables move in the same direction together.

36. Identify the $x$-values where the function is decreasing: points $(0,1)$, $(2,3)$, $(4,2)$, $(6,5)$ are connected in order.

    Decreasing on $2<x<4$. $y$ drops from $3$ to $2$ as $x$ goes from $2$ to $4$.

37. What is the qualitative meaning of a negative slope on a graph of $y$ versus $x$?

    $y$ decreases as $x$ increases. Negative slope means downward trend from left to right.

38. Identify the sign of the slope if a line passes through $(0,2)$ and $(4,6)$.

    Positive slope, since $6>2$ when $4>0$. $y$ increases from $2$ to $6$ as $x$ increases.

39. What does the slope represent for a graph of $y$ versus $x$?

    The rate of change: $\frac{\text{change in }y}{\text{change in }x}$. Slope measures steepness and direction of change.

40. What does it mean if the slope is getting larger as $x$ increases (line becomes steeper)?

    The rate of change is increasing (not constant). Steeper slopes indicate faster rates of change.

41. Identify the qualitative graph: starts at $(0,0)$, increases fast, then levels off as $x$ increases.

    Increasing and nonlinear; slope decreases toward $0$. Curve flattens as growth slows down.

42. Identify the sign of the slope if a line passes through $(1,5)$ and $(3,1)$.

    Negative slope, since $1<5$ when $3>1$. $y$ decreases from $5$ to $1$ as $x$ increases.

43. Which sketch matches: increasing at a constant rate with $y$-intercept $2$?

    An upward straight line crossing the $y$-axis at $(0,2)$. Constant positive slope starting at $y=2$.

44. What does it mean on a graph when a function is decreasing as $x$ increases?

    As $x$ increases, $y$ decreases (graph falls left to right). Falling graphs show negative relationships between variables.

45. Which is steeper: line $A$ with slope $2$ or line $B$ with slope $\frac{1}{2}$?

    Line $A$ is steeper because $|2|>|\frac{1}{2}|$. Compare absolute values of slopes for steepness.

46. What is the $x$-intercept on a graph, in words and notation?

    The value of $x$ when $y=0$; point $(a,0)$. Where the graph crosses the horizontal axis.

47. Which sketch description matches: "Starts high, decreases quickly, then levels off"?

    A decreasing curve that becomes flatter as $x$ increases. Shows exponential decay behavior.

48. What is the slope of the line through $(-1,4)$ and $(3,0)$?

    $m=\frac{0-4}{3-(-1)}=-1$. Apply slope formula with given points.

49. What does it mean on a graph when a function is increasing as $x$ increases?

    As $x$ increases, $y$ increases (graph rises left to right). Rising graphs show positive relationships between variables.

50. What is the $y$-intercept on a graph, in words and notation?

    The value of $y$ when $x=0$; point $(0,b)$. Where the graph crosses the vertical axis.

51. What qualitative graph matches "decreases at a constant rate"?

    A straight line falling left to right (linear with $m<0$). Constant negative slope creates steady downward line.

52. Identify the intervals of increase for a V-shape with vertex at $(0,0)$ opening upward.

    Increasing for $x>0$ (and decreasing for $x<0$). V-shape rises on the right side of vertex.

53. Identify whether the function is increasing, decreasing, or constant if its line has slope $m=0$.

    Constant. Zero slope produces a horizontal line.

54. What is the qualitative meaning of a slope of $0$ on a graph of $y$ versus $x$?

    $y$ is constant as $x$ changes. Zero slope means a horizontal line.

55. What is the slope interpretation when a line falls from left to right?

    Negative slope: $m<0$. Falling lines have negative slopes by definition.

56. What does it mean for a function to be constant on an interval?

    As $x$ changes, $y$ stays the same on that interval. The function has zero slope on that interval.

57. What graphical feature shows that a relationship is linear?

    The graph is a straight line with constant slope. Linear means constant rate of change between any two points.

58. What graph feature shows a function is linear rather than nonlinear?

    A linear function graphs as a straight line. Linear functions have constant rates of change.

59. What feature tells you a relationship is nonlinear when you look at its graph?

    The graph is not a straight line (it is curved or changes slope). Nonlinear functions have varying rates of change.

60. What feature tells you a relationship is linear when you look at its graph?

    The graph is a straight line. Linear functions have constant rate of change.

61. What does it mean if a graph becomes steeper as $x$ increases?

    The rate of change is increasing; $y$ changes faster per $1$ in $x$. Steeper slope means faster rate of change.

62. Which option describes a linear graph: A) curve B) straight line C) zigzag with corners?

    B) Straight line. Linear graphs are always straight lines.

63. Identify the slope when a graph is horizontal.

    Zero slope: $m=0$. Horizontal lines have no rise, only run.

64. What does a slope of $0$ mean about the graph and the relationship?

    The graph is horizontal; $y$ is constant as $x$ changes. No slope means no vertical change.

65. Which description matches a line with equation $y=\frac{1}{2}x-7$: increasing or decreasing?

    Increasing, because the slope is $\frac{1}{2}$. Positive slope means the line rises.

66. What does it mean for a function to be decreasing on an interval in a graph?

    As $x$ increases, $y$ decreases (the graph falls left to right). Falling graph means negative change in $y$ as $x$ grows.

67. Identify the $y$-intercept of the line through $(0,-2)$ and $(3,4)$.

    $y$-intercept is $-2$ (point $(0,-2)$). The line passes through $(0,-2)$.

68. What graphical feature shows that a relationship is nonlinear?

    The graph is not a straight line; the slope changes. Curves or bends indicate varying rates of change.

69. Which interval is increasing if a graph rises from $x=1$ to $x=4$?

    Increasing on $1<x<4$. Graph rises between these $x$-values.

70. Which interval is decreasing if a graph falls from $x=-2$ to $x=3$?

    Decreasing on $-2<x<3$. Graph falls between these $x$-values.

71. Which description matches a line with equation $y=-3x+2$: increasing or decreasing?

    Decreasing, because the slope is $-3$. Negative slope means the line falls.

72. What is an $x$-intercept on a graph, in terms of $x$ and $y$?

    A point where $y=0$; it is $(a,0)$. The $x$-intercept occurs where the graph crosses the $x$-axis.

73. What qualitative change occurs if a graph switches from falling to rising at one point?

    It changes from decreasing to increasing at a turning point (minimum). A valley where direction reverses.

74. Is the line through $(0,3)$ and $(4,1)$ increasing or decreasing?

    Decreasing, since $m=\frac{1-3}{4-0}=-\frac{1}{2}$. Negative slope means the line falls.

75. Which interval shows the function increasing for points $(1,2)$, $(3,5)$, $(6,4)$?

    Increasing on $1\le x\le 3$. $y$ goes from 2 to 5 as $x$ goes from 1 to 3.

Study these flashcards