Compare Functions in Different Representations - 8th Grade Math

1. Which statement best compares slopes: $m=-2$ versus $m=1$?

   $1$ is greater than $-2$. Positive slopes indicate increasing functions; negative slopes indicate decreasing.

2. Identify the missing value if a linear table has constant rate $3$: $(0,2)$, $(1,5)$, $(2,?)$.

   $8$. With rate $3$, each $x$ increase of $1$ adds $3$ to $y$: $5+3=8$.

3. Identify the rate of change for a function described as "$y$ decreases $5$ for each increase of $1$ in $x$".

   $-5$. A decrease of $5$ per unit increase means the slope is negative: $-5$.

4. Which function has the greater rate of change: $g(x)=-x+6$ or table $(1,2)$, $(4,8)$?

   The table function. $g$ has slope $-1$; table has slope $\frac{8-2}{4-1}=2$.

5. Which table shows the greater rate of change: A has $(0,1),(2,5)$; B has $(0,1),(2,3)$?

   Table A. Table A: slope $=\frac{5-1}{2-0}=2$; Table B: slope $=\frac{3-1}{2-0}=1$.

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

   Undefined. Vertical lines have zero run in the slope formula.

7. What is the slope of the function $y=-3x+5$?

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

8. What is the slope of a vertical line such as $x=2$?

   Undefined. Vertical lines have no horizontal change, making slope division by zero.

9. What is the rate of change from the table points $ (1,5) $ and $ (3,9) $?

   $2$. Using slope formula: $\frac{9-5}{3-1}=\frac{4}{2}=2$.

10. Which function is steeper: $y=\frac{1}{2}x+1$ or $y=-\frac{3}{4}x+2$?

    $y=-\frac{3}{4}x+2$. Compare absolute values: $|\frac{1}{2}|<|\frac{3}{4}|$.

11. What formula finds rate of change between points $(x_1,y_1)$ and $(x_2,y_2)$?

    $\frac{y_2-y_1}{x_2-x_1}$. This formula calculates slope as rise over run between two points.

12. A tank starts with $50$ L and drains $3$ L per minute. What is the initial value?

    $50$. The tank contains $50$ L at time $t=0$.

13. What is the slope of a line passing through $(0,3)$ and $(2,7)$?

    $2$. Using slope formula: $\frac{7-3}{2-0}=\frac{4}{2}=2$.

14. Which function has the greater initial value: $f(x)=3x-2$ or $g(x)=3x+5$?

    $g(x)=3x+5$. Compare y-intercepts: $-2<5$.

15. Identify the rate of change for $f(x)=\frac{2}{3}x-4$.

    $\frac{2}{3}$. The coefficient of $x$ gives the rate of change.

16. What is the initial value of the function given by the points $(0,7)$ and $(4,15)$?

    $7$. The y-value when $x=0$ is the initial value.

17. What is the slope of a horizontal line given by $y=7$?

    $0$. Horizontal lines have no vertical change.

18. Which function is increasing: $f(x)=- \frac{1}{2}x+3$ or $g(x)= \frac{3}{4}x-2$?

    $g(x)$. $g(x)$ has positive slope $\frac{3}{4}$, while $f(x)$ has negative slope $-\frac{1}{2}$.

19. What property of a linear function equals its rate of change in $y=mx+b$?

    The slope $m$. In $y=mx+b$, the coefficient $m$ represents how much $y$ changes per unit change in $x$.

20. Which function has the greater rate of change: $f(x)=2x-1$ or table $(0,0)$, $(2,3)$?

    $f(x)=2x-1$. $f$ has slope $2$; table has slope $\frac{3-0}{2-0}=1.5$.

21. Which value is the initial value in $y=mx+b$, and what does it mean?

    $b$; the $y$-intercept (value when $x=0$). The constant term shows where the line crosses the y-axis.

22. Which function has greater slope: line through $(0,0)$ and $(4,8)$ or $y= \frac{3}{2}x+1$?

    The line through $(0,0)$ and $(4,8)$. Line has slope $\frac{8-0}{4-0}=2$, greater than $\frac{3}{2}$ from the equation.

23. Choose the correct comparison: slopes $m=\frac{5}{2}$ and $n=2.4$; which is greater?

    $m=\frac{5}{2}$. $\frac{5}{2}=2.5>2.4$, so $m$ is greater.

24. Which function decreases faster: $f(x)=-4x+2$ or $g(x)=-x-10$?

    $f(x)$. $f(x)$ has slope $-4$ (more negative) versus $g(x)$ with slope $-1$.

25. Which representation shows a constant rate of change: a straight-line graph or a curved graph?

    A straight-line graph. Linear functions have constant slope, appearing as straight lines on graphs.

26. What is the slope of a horizontal line (constant function) such as $y=6$?

    $0$. Horizontal lines have no vertical change, so slope equals zero.

27. Identify the initial value for a function described as "When $x=0$, $y=12$".

    $12$. The initial value is the $y$-value when $x=0$.

28. What is the initial value (starting value) of the function $y=4x-9$?

    $-9$. The initial value is the $y$-intercept, found when $x=0$: $y=4(0)-9=-9$.

29. What does the slope $m$ represent in a linear function $y=mx+b$?

    The rate of change (change in $y$ for each $1$ change in $x$). Slope measures how steep the line is.

30. What is the rate of change from the table points $(1,4)$ and $(3,10)$?

    $3$. Use $m=\frac{10-4}{3-1}=\frac{6}{2}=3$.

31. A taxi charges $4 to start plus $2 per mile. What is the rate of change?

    $2$ dollars per mile. Cost increases by $2 for each mile traveled.

32. Which slope is greater: $m_1=-2$ or $m_2=-\frac{1}{2}$?

    $m_2=-\frac{1}{2}$. $-\frac{1}{2}$ is closer to zero, so it's greater.

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

    $m=\frac{y_2-y_1}{x_2-x_1}$. Rise over run formula for finding slope.

34. What is the rate of change of the function $y=-3x+7$?

    $-3$. The coefficient of $x$ in $y=mx+b$ form gives the rate of change.

35. Identify whether the function is increasing, decreasing, or constant if $m<0$.

    Decreasing. Negative slopes go down from left to right.

36. Identify whether the function is increasing, decreasing, or constant if $m>0$.

    Increasing. Positive slopes go up from left to right.

37. Which function has the greater rate of change: $f(x)=2x+1$ or $g(x)=-x+5$?

    $f(x)$. $f(x)$ has slope $2$ while $g(x)$ has slope $-1$, so $f(x)$ changes faster.

38. Which is larger: the rate of change $\frac{1}{3}$ or $\frac{2}{5}$?

    $\frac{2}{5}$. Converting to decimals: $\frac{1}{3} \approx 0.33$ and $\frac{2}{5} = 0.4$.

39. Which function has the greater initial value: $f(x)=3x-4$ or $g(x)=-2x+1$?

    $g(x)$. $f(0)=-4$ and $g(0)=1$, so $g(x)$ has the greater initial value.

40. What is the rate of change from the table points $(0,-1)$ and $(2,5)$?

    $3$. Use $m=\frac{5-(-1)}{2-0}=\frac{6}{2}=3$.

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