Interpret Linear Functions - International Brotherhood of Electrical Workers Exam IBEW

1. What is the $y$-intercept of the line graphed by the function $y=-\frac{1}{2}x+3$?

   $b=3$. In slope-intercept form, the constant term is the y-intercept.

2. Identify the $y$-intercept of the line passing through $(2,1)$ with slope $m=4$.

   $b=-7$. Using point-slope form y - 1 = 4(x - 2) simplifies to y = 4x - 7, where -7 is the y-intercept.

3. Identify whether the table is linear: $(0,2)$, $(1,5)$, $(2,9)$, $(3,14)$.

   Not linear (differences in $y$ are $3,4,5$). Linear functions require constant differences in y for equal x-intervals, but here they vary (3,4,5).

4. What is the value of $y$ when $x=5$ for the linear function with $m=-2$ and $b=9$?

   $y=-1$. Substitute x=5 into y = -2x + 9 to get y = -10 + 9 = -1.

5. Identify the equation of the line that crosses the $y$-axis at $5$ and rises $2$ for every run of $1$.

   $y=2x+5$. The description matches slope-intercept form with m=2 and b=5.

6. What is the $x$-intercept of a line in terms of $m$ and $b$ for $y=mx+b$?

   $x=-\frac{b}{m}$ (when $m\neq 0$). Found by setting y=0 in the equation and solving for x, yielding the point where the line crosses the x-axis.

7. What does the $y$-intercept $b$ represent in $y=mx+b$?

   The value of $y$ when $x=0$. Represents the point where the line crosses the y-axis, found by setting x to zero in the equation.

8. Identify the slope of the line given by $3x-2y=8$.

   $m=\frac{3}{2}$. Rewrite the equation in slope-intercept form y = (3/2)x - 4, where the coefficient of x is the slope.

9. Identify the $y$-intercept of the line given by $3x-2y=8$.

   $b=-4$. From the slope-intercept form y = (3/2)x - 4, the constant term is the y-intercept.

10. What is the slope if a line passes through $(2,5)$ and $(6,13)$?

    $m=2$. Apply the slope formula: (13-5)/(6-2) = 8/4 = 2, showing the rate of change.

11. What is the slope of a line that passes through $(0,3)$ and $(6,3)$?

    $m=0$. The y-values are constant, so change in y is 0, yielding slope (3-3)/(6-0) = 0.

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

    Rate of change: change in $y$ per $1$ unit change in $x$. Quantifies the constant rate at which y changes for each unit increase in x in a linear relationship.

13. What is the slope of the line through $(0,-1)$ and $(4,7)$?

    $m=2$. Using the slope formula: (7 - (-1))/(4 - 0) = 8/4 = 2, which is the constant rate of change.

14. What is the slope of a vertical line (equation $x=c$)?

    Slope is undefined. Vertical lines have zero horizontal change, making the slope formula division by zero.

15. What is the slope between points $( -2,4)$ and $(1,-2)$?

    $m=-2$. Slope formula gives (-2 - 4)/(1 - (-2)) = -6/3 = -2.

16. Identify the constant rate of change from the table: $(1,4)$, $(2,7)$, $(3,10)$, $(4,13)$.

    $m=3$. The y-values increase by 3 for each unit increase in x, indicating a constant slope of 3.

17. Identify the slope of the line graphed by the function $y=-\frac{1}{2}x+3$.

    $m=-\frac{1}{2}$. In slope-intercept form, the coefficient of x is the slope.

18. What is the equation of the line with slope $m=3$ and $y$-intercept $b=-2$?

    $y=3x-2$. In slope-intercept form, substitute the given slope as the coefficient of x and the y-intercept as the constant.

19. Find the $y$-intercept from the table: $(1,4)$, $(2,7)$, $(3,10)$, $(4,13)$.

    $b=1$. Extend the pattern backward: when x=0, y=1, as each step adds 3 starting from there.

20. Identify whether the function is increasing if its slope is $m=-4$.

    No; it is decreasing. A negative slope indicates that as x increases, y decreases, defining a decreasing function.

21. What is the slope from a table if $x$ increases by $2$ and $y$ increases by $-6$ each step?

    $m=-3$. Slope is the ratio of change in y to change in x, so -6/2 = -3 for each step.

22. What is the $y$-intercept of the line through $(0,-1)$ and $(4,7)$?

    $b=-1$. The line passes through (0, -1), so when x=0, y=-1, which is the y-intercept.

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

    $m=\frac{y_2-y_1}{x_2-x_1}$. Calculates the ratio of vertical change to horizontal change between two points on a line.

24. What is the slope of the line shown by points $(1,2)$ and $(3,6)$ on a graph?

    $m=2$. Apply the slope formula: (6-2)/(3-1) = 4/2 = 2, representing rise over run.

25. What is the slope of a horizontal line (equation $y=c$)?

    $m=0$. Horizontal lines have no vertical change, resulting in zero slope regardless of horizontal distance.

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