Construct and Interpret Linear Functions - 8th Grade Math

1. Interpret $m=-3$ in $y=-3x+10$ for a real situation where $x$ is hours and $y$ is dollars.

   dollars decrease by $3$ each hour. Negative slope means decreasing relationship.

2. What is the meaning of a slope of $0$ for a linear relationship?

   $y$ stays constant as $x$ changes. Zero slope creates a horizontal line.

3. Identify the slope and initial value from the statement: "You earn $12 per hour plus a $30 bonus."

   Rate = $12$, initial = $30$. Hourly rate is slope; bonus is initial value.

4. Interpret $b=8$ in $y=1.5x+8$ when $x$ is minutes and $y$ is pages read.

   at $x=0$, $y=8$ pages already read. The $y$-intercept shows the starting point before timing begins.

5. From the table points $(1,4)$ and $(4,13)$, what is the rate of change?

   $m=3$. $m=\frac{13-4}{4-1}=\frac{9}{3}=3$

6. What is the slope (rate of change) 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: change in $y$ divided by change in $x$.

7. Which equation form shows a linear function using slope and initial value?

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

8. A line has slope $m = 5$ and passes through $(0,-1)$. What is its equation?

   $y = 5x - 1$. When line passes through $(0,b)$, the equation is $y = mx + b$.

9. A graph shows points $(0,2)$ and $(4,10)$. What is the slope $m$?

   $m = 2$. Calculate $m = \frac{10-2}{4-0} = \frac{8}{4} = 2$.

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

    $y$ decreases as $x$ increases. Negative slope means the line goes down from left to right.

11. Find $b$ for the line with $m=3$ that passes through $(2,7)$.

    $b=1$. Substitute into $7=3(2)+b$ to get $b=1$.

12. In $C(m) = 2m + 3$, what does the $2$ represent in the taxi situation?

    $2 per mile (rate of change). The coefficient of $m$ represents cost per mile.

13. In $C(m) = 2m + 3$, what does the $3$ represent in the taxi situation?

    $3 starting fee (initial value). The constant term represents the fixed initial cost.

14. Write the function rule in slope-intercept form through $(0,4)$ and $(3,10)$.

    $y=2x+4$. $m=\frac{10-4}{3-0}=2$, and $b=4$ from point $(0,4)$.

15. A tank has $50$ liters and drains $4$ liters per minute. What is $V(t)$ after $t$ minutes?

    $V(t)=50-4t$. Initial volume minus drain rate times time.

16. Using the table $x: 0,2,4$ and $y: 3,7,11$, what is the initial value?

    $b=3$. When $x=0$, $y=3$ is the initial value.

17. Identify the slope and initial value in $y = -4x + 7$.

    $m = -4, b = 7$. Read directly: coefficient of $x$ is slope, constant is initial value.

18. Identify the initial value for the function $y=4x-9$.

    $b=-9$. The constant term in slope-intercept form is the initial value.

19. A tank has $50$ L and drains $3$ L per minute. What is $V(t)$ after $t$ minutes?

    $V(t) = 50 - 3t$. Volume decreases by 3L per minute from initial 50L.

20. What is the slope of the line through $(2,5)$ and $(6,13)$?

    $m = 2$. Use $m = \frac{13-5}{6-2} = \frac{8}{4} = 2$.

21. Write the linear function for a line with slope $m=-2$ and $y$-intercept $b=6$.

    $y=-2x+6$. Substitute slope and $y$-intercept into $y=mx+b$.

22. What is the slope-intercept form of a linear function?

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

23. Find the initial value $b$ if $m = 3$ and the line passes through $(2,11)$.

    $b = 5$. Substitute into $11 = 3(2) + b$ to get $b = 5$.

24. Identify the $y$-intercept on a graph in terms of coordinates.

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

25. What does $b$ represent in $y=mx+b$ in a real-world situation?

    initial value (starting amount) when $x=0$. The $y$-intercept represents the starting value before any change occurs.

26. What is the meaning of a negative slope for a linear relationship?

    As $x$ increases, $y$ decreases at a constant rate. Negative slope means the line goes down from left to right.

27. Find the function rule for the linear relationship with points $(3,-1)$ and $(7,11)$.

    $y=3x-10$. Slope is $\frac{11-(-1)}{7-3}=3$; use point to find $b=-10$.

28. What is the formula for slope (rate of change) between $(x_1,y_1)$ and $(x_2,y_2)$?

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

29. For $V(t) = 50 - 3t$, identify the rate of change and initial value.

    Rate $=-3$ L/min, initial $=50$ L. Negative slope shows decrease; constant is starting volume.

30. A table shows $x: 0,2,4$ and $y: 3,7,11$. What is the rate of change?

    $m=2$. $y$ increases by $4$ when $x$ increases by $2$, so $m=2$.

31. Construct the linear function through $(1,2)$ and $(3,8)$ in $y = mx + b$ form.

    $y = 3x - 1$. Find $m = \frac{8-2}{3-1} = 3$, then use point to find $b = -1$.

32. A taxi costs $3 to start plus $2 per mile. What is the function $C(m)$?

    $C(m) = 2m + 3$. Cost equals rate per mile times miles plus fixed starting fee.

33. Find $b$ for a line with slope $m=3$ that passes through $(2,11)$.

    $b=5$. Use $11=3(2)+b$, so $11=6+b$, thus $b=5$.

34. A taxi charges a $5 start fee plus $2 per mile. What is the function $C(m)$?

    $C(m)=2m+5$. Fixed fee is initial value, per-mile rate is slope.

35. What does $m$ represent in $y=mx+b$ in a real-world situation?

    rate of change per $1$ unit increase in $x$. Slope tells how much $y$ changes for each unit increase in $x$.

36. Choose the correct function for "Starts at $8$ and increases by $\frac{1}{2}$ each step" with input $x$.

    $y = \frac{1}{2}x + 8$. Starting value is $b=8$; rate of increase is $m=\frac{1}{2}$.

37. A runner's distance is $d=6t+2$ (miles). What does the $2$ mean in context?

    The runner started $2$ miles from the starting point at $t=0$. The constant term represents starting position.

38. In $y=mx+b$, what does $b$ represent in a real-world situation?

    $b$ is the initial value (the $y$-value when $x=0$). The $y$-intercept is the starting value when input is zero.

39. What does the initial value $b$ represent in $y = mx + b$?

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

40. A line passes through $(0,-2)$ and $(4,6)$. What is the linear function $y=$?

    $y=2x-2$. $m=\frac{6-(-2)}{4-0}=2$; $y$-intercept is $-2$.

41. A tank has $50$ liters and drains $4$ liters per minute. What is the function $V(t)$?

    $V(t)=-4t+50$. Starting volume minus drain rate times minutes.

42. Which point on a graph gives the initial value of a linear function?

    The $y$-intercept, $(0,b)$. When $x=0$, $y=b$, which is the initial value.

43. Identify the slope of the line given by $y=-3x+7$.

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

44. Construct the linear function with slope $m = -2$ and point $(4,1)$.

    $y = -2x + 9$. Substitute $(4,1)$ into $y = -2x + b$ to find $b = 9$.

45. Find the slope of the line through $(2,5)$ and $(6,13)$.

    $m=2$. $m=\frac{13-5}{6-2}=\frac{8}{4}=2$

46. What is the rate of change from the table where $(x,y)$ includes $(0,4)$ and $(2,10)$?

    $m = 3$. Calculate $m = \frac{10-4}{2-0} = \frac{6}{2} = 3$.

47. What is the initial value from the table with points $(0,-2)$ and $(3,4)$?

    $b = -2$. The y-value at $x=0$ is the initial value.

48. A line crosses the $y$-axis at $(0,-6)$. What is the initial value $b$?

    $b = -6$. The y-coordinate of the y-intercept is the initial value.

49. What is the $y$-intercept of a linear function in terms of a point on its graph?

    The point where $x=0$, written as $(0,b)$. The $y$-intercept occurs where the line crosses the $y$-axis.

50. A taxi charges a $3 start fee plus $2 per mile. What is the function for cost $C$ vs. miles $m$?

    $C=2m+3$. Fixed fee is initial value, per-mile rate is slope.

51. A gym membership costs $20$ to join and $15$ per month. What is the initial value in this model?

    $20$. The joining fee is the cost when months = 0.

52. On a graph, a line crosses the $y$-axis at $(0,5)$ and rises $2$ for every run of $1$. What is $y=$?

    $y=2x+5$. Rise of $2$ over run of $1$ gives slope $m=2$; crosses at $(0,5)$ so $b=5$.

53. Write the function rule through $(1,2)$ and $(3,8)$ in slope-intercept form.

    $y=3x-1$. $m=\frac{8-2}{3-1}=3$; use point-slope to find $b=-1$.

54. What is the slope of a horizontal line (constant function) $y=c$?

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

55. In $y=mx+b$, what does $m$ represent in a real-world situation?

    $m$ is the rate of change (change in $y$ per $1$ unit of $x$). Slope tells how much $y$ changes for each unit increase in $x$.

56. Write the equation of the line passing through $ (0,-4) $ and $ (5,1) $.

    $y=x-4$. Slope is $\frac{1-(-4)}{5-0}=1$, and $b=-4$ from $ (0,-4) $.

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