Identify Linear Functions - International Brotherhood of Electrical Workers Exam IBEW

1. Identify whether $x^2+y^2=9$ represents a linear function (in the plane).

   Not linear (it is a circle). Squared variables produce a circular relation, not a straight line characteristic of linearity.

2. Identify whether points $(0,2)$, $(2,6)$, $(4,10)$ lie on a linear function.

   Linear. Uniform slope of 2 between points indicates they are collinear on a straight line.

3. Identify whether $3x-2y=8$ represents a linear function (in the plane).

   Linear relation (a line). The equation $ax+by=c$ in two variables always graphs as a straight line in the plane.

4. Identify whether $f(x)=|x|+1$ is linear.

   Not linear. Absolute value introduces a kink at $x=0$, preventing a single constant slope.

5. Identify whether the table is linear: $(1,3)$, $(2,5)$, $(3,7)$, $(4,9)$.

   Linear. Consistent y-differences of 2 for x-increments of 1 yield a constant slope.

6. Which equation is linear in $x$: $2x+3y=6$ or $2x^2+3y=6$?

   $2x+3y=6$. Solving for y gives $y=-\frac{2}{3}x+2$, which is in linear form without higher powers.

7. Identify whether $f(x)=\frac{4}{3}x+2$ is linear.

   Linear. The equation follows the form $f(x)=mx+b$ with fractional slope, maintaining constant change.

8. Which option indicates a non-linear function: $y=mx+b$ or $y=ax^2+bx+c$?

   $y=ax^2+bx+c$. The $ax^2$ term creates a parabola, deviating from the straight line of linear functions.

9. Identify whether $f(x)=3(x-2)+5$ is linear.

   Linear. Distributing simplifies to $3x-1$, conforming to the linear form $mx+b$.

10. Identify whether $f(x)=\frac{2x-1}{x+4}$ is linear.

    Not linear. The rational expression does not simplify to $mx+b$ due to $x$ in the denominator.

11. What is the standard form of a linear function in one variable?

    $f(x)=mx+b$. This form captures all linear functions with slope $m$ and y-intercept $b$, ensuring a constant rate of change.

12. Identify whether $f(x)=7$ is linear.

    Linear (constant function; slope $m=0$). Constant functions are linear with zero slope, expressible as $f(x)=0 cdot x + c$.

13. Which equation form is linear: $y=3x-5$ or $y=3x^2-5$?

    $y=3x-5$. It matches the slope-intercept form $y=mx+b$ without higher powers or other non-linear terms.

14. Identify whether $f(x)=\frac{1}{2}x^2+3$ is linear.

    Not linear. The $x^2$ term introduces curvature, making the graph parabolic rather than straight.

15. Identify whether $f(x)=\frac{4}{x}+2$ is linear.

    Not linear. The presence of $x$ in the denominator creates a hyperbolic curve, violating linearity.

16. What condition on the rate of change determines that a function is linear?

    Constant slope (constant rate of change). Linearity requires the rate of change to remain constant, producing a uniform slope throughout.

17. Identify whether $f(x)=\frac{2x-1}{4}$ is linear.

    Linear. Simplifying yields $\frac{1}{2}x - \frac{1}{4}$, which fits the linear form $mx+b$.

18. Identify whether points $(1,2)$, $(2,4)$, $(3,7)$ lie on a linear function.

    Not linear. Inconsistent slopes (2 and 3) show the points are not collinear, hence non-linear.

19. Identify whether the graph shown is linear (points $(0,1)$, $(1,3)$, $(2,5)$).

    Linear. Constant slope of 2 between consecutive points confirms a linear relationship.

20. Identify whether the table is linear: $(0,0)$, $(1,1)$, $(2,4)$, $(3,9)$.

    Not linear. Increasing y-differences (1,3,5) indicate a non-constant rate of change, typical of quadratics.

21. What is the defining graphical feature of a linear function?

    Its graph is a straight line. A constant rate of change results in a graph that forms a straight line without curves or bends.

22. Identify whether $f(x)=(x-2)(x+5)$ is linear.

    Not linear. Expanding produces $x^2+3x-10$, with an $x^2$ term indicating quadratic nature.

23. What is the slope-intercept form criterion for linearity in $x$?

    Only $x^1$ appears; no $x^2$, $x$ in denominators, or $|x|$. Linearity in slope-intercept form requires only first-degree $x$ without non-linear operations.

24. What test using two points determines if a relation is linear?

    Slope between any two points is constant. Consistency in slope across all point pairs confirms collinearity and thus linearity.

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