Interpreting Graphs and Tables - ISEE Middle Level Quantitative Reasoning

1. What does the $x$-axis represent on most graphs?

   The independent variable (input). In standard coordinate graphs, the horizontal x-axis displays the independent variable, which is manipulated or input to observe changes in the dependent variable.

2. A table lists $x: 2,4,6$ and $y: 5,10,15$. What is the constant of proportionality $k$ in $y=kx$?

   $\frac{5}{2}$. Dividing $y$ by $x$ for each pair yields $5/2 = 10/4 = 15/6 = \frac{5}{2}$, the constant $k$ in direct proportion.

3. What does a negative slope indicate about how $y$ changes as $x$ increases?

   $y$ decreases as $x$ increases. A negative slope means the line falls from left to right, indicating that the dependent variable decreases as the independent variable increases.

4. What is the $y$-intercept of a graph in terms of an ordered pair?

   The point where $x=0$. The y-intercept is the point where the graph crosses the y-axis, occurring when the independent variable $x$ is zero.

5. Find the slope between $(1,7)$ and $(4,1)$.

   $-2$. Applying the slope formula gives $(1-7)/(4-1) = -6/3 = -2$, showing the negative rate of change.

6. Identify the slope of a horizontal line on a graph.

   Slope $0$. A horizontal line has no vertical change, resulting in a slope of zero when applying the slope formula.

7. A line graph shows a value of $20$ at $t=2$ and $26$ at $t=5$. What is the average change per unit $t$?

   $2$ per unit $t$. The average rate is the total change in value divided by the change in $t$, so $(26-20)/(5-2) = 6/3 = 2$.

8. A table shows $(x,y)$ pairs: $(0,10)$, $(1,7)$, $(2,4)$. What is the rate of change?

   $-3$ per $1$ unit of $x$. The consistent decrease of 3 in $y$ for each increase of 1 in $x$ indicates a constant rate of change of -3.

9. What does a positive slope indicate about how $y$ changes as $x$ increases?

   $y$ increases as $x$ increases. A positive slope means the line rises from left to right, showing that the dependent variable increases with the independent variable.

10. A line passes through $(0,3)$ and $(2,7)$. What is its slope?

    $2$. The slope is calculated as $(7-3)/(2-0) = 4/2 = 2$, representing the constant rate of change for the line.

11. A line has slope $-3$ and passes through $(0,4)$. What is $y$ when $x=2$?

    $-2$. Using $y = -3x + 4$, substitute $x=2$ to get $y = -6 + 4 = -2$, following the line equation.

12. A bar chart shows Category A $=12$ and Category B $=18$. What is the difference $B-A$?

    $6$. Subtracting the values directly from the bar chart gives $18 - 12 = 6$, representing the difference between categories.

13. Identify the slope of a vertical line on a graph.

    Undefined slope. A vertical line has no horizontal change, leading to division by zero in the slope formula, which is undefined.

14. What is the meaning of the ordered pair $(x,y)$ on a coordinate graph?

    At input $x$, the output is $y$. An ordered pair $(x,y)$ indicates a specific point where the input value $x$ corresponds to the output value $y$ in the relationship being graphed.

15. A pie chart shows $40\%$ of $150$ items are defective. How many are defective?

    $60$. Converting $40\%$ to 0.4 and multiplying by 150 gives $0.4 \times 150 = 60$, quantifying the defective items.

16. Find the slope between $(2,5)$ and $(6,13)$.

    $2$. Using the slope formula, $(13-5)/(6-2) = 8/4 = 2$, confirming the rate of change between the points.

17. Identify whether the relationship is proportional if points include $(0,3)$ and $(2,6)$.

    Not proportional (does not pass through $(0,0)$). A proportional relationship requires passing through (0,0), but here y=3 at x=0, so it is not proportional.

18. In a table, $y$ increases by $15$ when $x$ increases by $3$. What is the unit rate?

    $5$ per $1$ unit of $x$. The unit rate is the change in $y$ divided by the change in $x$, so $15/3=5$ per unit increase in $x$.

19. A table shows $(x,y)$ pairs: $(1,4)$, $(2,6)$, $(3,8)$. What is $y$ when $x=5$?

    $12$. The pattern shows a linear relationship with slope 2 and y-intercept 2, so for $x=5$, $y=2(5)+2=12$.

20. What is the slope between points $(x_1,y_1)$ and $(x_2,y_2)$?

    $m=\frac{y_2-y_1}{x_2-x_1}$. The slope formula calculates the rate of change as the difference in $y$-values divided by the difference in $x$-values between two points.

21. What does the $y$-axis represent on most graphs?

    The dependent variable (output). The vertical y-axis on graphs typically plots the dependent variable, which responds to or is affected by changes in the independent variable.

22. What is the $x$-intercept of a graph in terms of an ordered pair?

    The point where $y=0$. The x-intercept is the point where the graph crosses the x-axis, occurring when the dependent variable $y$ is zero.

23. A pie chart shows $25\%$ of a group of $80$ students. How many students is that?

    $20$ students. Calculating $25\%$ of 80 as $0.25 \times 80 = 20$ determines the portion represented in the pie chart.

24. Which option is the best estimate if a graph's scale is $5$ units per tick and the point is $3$ ticks up?

    $15$ units. Multiplying the number of ticks by the scale per tick gives $3 \times 5 = 15$, estimating the value on the graph.

25. A scatter plot shows points close to a rising line from left to right. What correlation is shown?

    Positive correlation. Points clustering around an upward-trending line indicate that as one variable increases, the other tends to increase, showing positive correlation.

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