Interpret Linear Function Equations - 8th Grade Math

1. What is the output when $x=3$ for the function $y = 2x - 1$?

   $y=5$. Substitute: $y = 2(3) - 1 = 6 - 1$.

2. What is the graph of any equation in the form $y = mx + b$?

   A straight line. Linear equations always produce straight lines.

3. What is the graph of any linear function $y=mx+b$?

   A straight line. Linear functions always produce straight lines when graphed.

4. What is the $y$-intercept of the line $y = 7x - 3$?

   $-3$. The constant term is the $y$-intercept $b$.

5. Identify whether $y = 7$ is linear.

   Yes; it is linear with $m=0$ and $b=7$. Horizontal lines are linear with zero slope.

6. What is the slope of the vertical line $x=-2$?

   Undefined slope. Vertical lines have infinite steepness, so slope is undefined.

7. Identify whether $A = s^2$ is linear or not linear.

   Not linear. Exponent makes it quadratic, not linear.

8. Identify the slope of the line given by $y=-3x+5$.

   $m=-3$. The coefficient of $x$ in slope-intercept form is the slope.

9. Identify whether points $(1,1)$, $(2,4)$, $(3,9)$ lie on one straight line.

   No, they are not collinear. Different slopes between pairs prove they're not on one line.

10. What is the constant rate of change (slope) for the points $ (1,5) $ and $ (4,11) $?

    2. Slope = $\frac{11-5}{4-1} = \frac{6}{3} = 2$

11. What is the constant rate of change for $y=4x-9$?

    $4$. The slope $m=4$ represents the constant rate of change.

12. Identify the $y$-intercept of the line given by $y = -3x + 5$.

    $b=5$. The constant term is the y-intercept.

13. What is the slope of the line through $ (0,2) $ and $ (3,11) $?

    $3$. Slope = $\frac{11-2}{3-0} = \frac{9}{3} = 3$

14. What is the value of $b$ in $y = mx + b$ if the line passes through $(0,-7)$?

    $-7$. Point $(0,b)$ is the $y$-intercept, so $b=-7$.

15. Which set of points is not on a straight line: $(1,1),(2,2),(3,3)$ or $(1,1),(2,4),(3,9)$?

    $(1,1),(2,4),(3,9)$. Second set forms parabola ($y=x^2$), not a line.

16. Which equation represents a linear function: $y = 3(x-1)$ or $y = (x-1)^2$?

    $y = 3(x-1)$. First simplifies to $y=-3x+3$ (linear); second is quadratic.

17. What is the slope of the line through $(-1,4)$ and $(2,-2)$?

    $-2$. Slope = $\frac{-2-4}{2-(-1)} = \frac{-6}{3} = -2$

18. What is the slope of the line through $(1,4)$ and $(5,4)$?

    $m=0$. Horizontal lines have zero slope since $y$ doesn't change.

19. What is the value of $x$ when $y=0$ for the function $y = 2x - 8$?

    $4$. Set $0=2x-8$; solving gives $2x=8$, so $x=4$.

20. What is the graph shape of any function written in the form $y = mx + b$?

    A straight line. Linear functions always produce straight lines when graphed.

21. Which equation is linear: $y = 2x + 1$ or $y = x^2 + 1$?

    $y = 2x + 1$. Only first-degree equations are linear.

22. Which equation represents a vertical line: $y=4$ or $x=4$?

    $x=4$. Vertical lines have constant $x$-values.

23. Which set of points is on a straight line: $(0,1),(1,3),(2,5)$ or $(1,1),(2,4),(3,9)$?

    $(0,1),(1,3),(2,5)$. Check if slope is constant between all pairs.

24. Which function is not linear because it has an exponent on the input: $A=s^2$ or $P=4s$?

    $A=s^2$. Squaring the input makes it nonlinear.

25. What is the slope of the horizontal line $y = 6$?

    $0$. Horizontal lines have no change in $y$, so slope is zero.

26. What does $b$ represent in $y=mx+b$?

    $b$ is the $y$-intercept (value when $x=0$). The point where the line crosses the y-axis.

27. What is the definition of a linear function in terms of rate of change?

    A function with a constant rate of change. The slope remains the same between any two points.

28. What is the slope of a horizontal line written as $y = 4$?

    $m=0$. Horizontal lines have no change in y.

29. Which equation is linear: $y = 3x + 1$ or $y = x^2 + 1$?

    $y = 3x + 1$. First has form $y=mx+b$; second has $x^2$ term.

30. Which equation is not linear: $y = 5x - 2$ or $y = \frac{3}{x}$?

    $y = \frac{3}{x}$. Division by x creates a curve, not a line.

31. Which equation represents a horizontal line: $y=4$ or $x=4$?

    $y=4$. Horizontal lines have constant $y$-values.

32. What is the slope formula using two points $(x_1,y_1)$ and $(x_2,y_2)$?

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

33. Identify whether $y=\frac{3}{x}$ is linear or not linear.

    Not linear. Rational functions with $x$ in denominator aren't linear.

34. What is the slope of the line through $(2,7)$ and $(6,19)$?

    $m=3$. $\frac{19-7}{6-2}=\frac{12}{4}=3$

35. Which equation has a constant rate of change: $y = -\frac{1}{2}x + 3$ or $y = x^2 - 1$?

    $y = -\frac{1}{2}x + 3$. First-degree equations have constant rates.

36. What is the function type of $A=s^2$ (area of a square) as a function of $s$?

    Not linear. The squared term creates a parabola, not a straight line.

37. What is the meaning of $m$ in the linear equation $y = mx + b$?

    $m$ is the slope (rate of change) of the line. It determines how steep the line is and its direction.

38. Which equation is not linear: $y = -2x + 9$ or $y = 5x^2$?

    $y = 5x^2$. Squared term makes it nonlinear; graph is a parabola.

39. What is the meaning of $b$ in the linear equation $y = mx + b$?

    $b$ is the $y$-intercept (the value of $y$ when $x=0$). It's where the line crosses the y-axis.

40. Which set has constant first differences: $2,5,8,11$ or $1,4,9,16$?

    $2,5,8,11$. Each term increases by 3, showing linear growth.

41. What is the slope of the vertical line $x = 2$?

    Undefined (not a function of $x$). Vertical lines have infinite steepness; $y$ isn't a function of $x$.

42. Identify whether the points $(0,0)$, $(2,4)$, and $(3,6)$ represent a linear relationship.

    Yes; the slope is constant ($m=2$). Calculate slopes: all equal $2$.

43. Find $b$ if the line has slope $m=2$ and passes through $(0,-5)$ in $y=mx+b$.

    $b=-5$. The y-intercept is the y-value at $x=0$.

44. What does $m$ represent in $y=mx+b$?

    $m$ is the slope (rate of change). It determines the steepness and direction of the line.

45. What is the value of $y$ when $x=4$ for the function $y = -3x + 2$?

    $-10$. Substitute: $y=-3(4)+2=-12+2=-10$.

46. Identify the slope of the line given by $y = -3x + 5$.

    $m=-3$. The coefficient of x is the slope.

47. What is the slope of the line $y = 7x - 3$?

    $7$. The coefficient of $x$ is the slope $m$.

48. Identify whether $y=x^2-1$ is linear or not linear.

    Not linear. The $x^2$ term makes it quadratic, not linear.

49. Identify the constant rate of change for the points $(1,2)$ and $(4,11)$.

    $m=\frac{11-2}{4-1}=3$. Use the slope formula with the given points.

50. What do the variables $m$ and $b$ represent in the linear equation $y = mx + b$?

    $m$ is slope; $b$ is the $y$-intercept. In slope-intercept form, $m$ controls steepness and $b$ is where line crosses $y$-axis.

51. Identify the slope and $y$-intercept in $y = -4x + 5$.

    $m=-4$, $b=5$. Coefficient of $x$ gives slope; constant gives $y$-intercept.

52. Identify whether $y = x^3$ is linear.

    No; it is not linear. Powers greater than 1 create curves.

53. What is the $y$-intercept point for the equation $y = mx + b$?

    $(0,b)$. When $x=0$, $y=b$ in the equation.

54. Identify whether $y=\sqrt{x}+2$ is linear or not linear.

    Not linear. Square root functions create curved graphs, not straight lines.

55. Identify whether $y=\frac{1}{2}x-7$ is linear.

    Linear. It's in the form $y=mx+b$ with $m=\frac{1}{2}$ and $b=-7$.

56. Identify the $y$-intercept of the line given by $y=-3x+5$.

    $b=5$. The constant term in slope-intercept form is the y-intercept.

57. What is the slope-intercept form of a linear equation?

    $y = mx + b$. Standard form showing slope $m$ and $y$-intercept $b$.

58. What is the $y$-intercept of the line $y = mx + b$ written as an ordered pair?

    $(0,b)$. When $x=0$, $y=m(0)+b=b$, giving the point $(0,b)$.

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