Identify Linear Functions - International Brotherhood of Electrical Workers Exam IBEW

Which of the following scenarios describes a relationship that is a linear function?
Answer: The total cost of buying spools of wire at $45 per spool. When you encounter questions about linear functions, you're looking for relationships where one variable changes at a constant rate with respect to another. Linear functions follow the form $y = mx + b$ , where the rate of change (slope) remains constant. Option A describes a linear function because the total cost increases by exactly $45 for each additional spool purchased. If you buy 1 spool, it costs $45; 2 spools cost $90; 3 spools cost $135, and so on. The relationship is $\text{Cost} = 45 \times \text{Number of spools}$ , which graphs as a straight line through the origin with a constant slope of 45. Option B is incorrect because the area of a circle follows $A = \pi r^2$ . Since the radius is squared, this creates a quadratic relationship where the rate of change accelerates as the radius increases. The graph would be a curved parabola, not a straight line. Option C is wrong because a thrown ball follows a parabolic path due to gravity. The height function is quadratic ( $h = -16t^2 + v_0t + h_0$ ), creating a curved trajectory where the ball slows down, stops, then accelerates downward. Option D represents exponential decay, following $A = A_0 \times (0.5)^t$ . The substance decreases by half each year, creating a curved exponential function rather than a straight line. Remember: linear functions have constant rates of change and graph as straight lines. Look for situations involving fixed unit costs, constant speeds, or steady increases/decreases per unit time.
A function is defined by the equation $y = 5$ . Is this a linear function?
Answer: Yes, it represents a horizontal line with a slope of zero. When you encounter questions about linear functions, focus on the fundamental definition: a linear function is any function that can be written in the form $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept. The key insight is that this includes cases where the slope equals zero. The equation $y = 5$ is indeed a linear function because it can be rewritten as $y = 0x + 5$ , which fits the standard linear form perfectly. This represents a horizontal line that crosses the y-axis at 5, with a slope of zero. Since the slope is zero, the line doesn't rise or fall—it remains constant at $y = 5$ for all x-values. Option A is incorrect because the value of x is irrelevant to whether this is linear—x can be any value while y remains 5. Option B reflects a common misconception that linear functions must explicitly show an x variable. However, $y = 5$ implicitly has an x term with coefficient zero ( $y = 0x + 5$ ). Option C confuses the definition of linear functions with the concept of variability. A constant output doesn't disqualify a function from being linear; it simply means the slope is zero. Option D correctly identifies this as a linear function representing a horizontal line with zero slope. Remember this pattern: constant functions like $y = k$ (where k is any number) are always linear functions—they're just horizontal lines. Don't let the absence of a visible x variable fool you into thinking they're not linear.
An electrician charges a $50 service fee plus $75 per hour of work. If C is the total cost and h is the number of hours worked, which statement is true?
Answer: The relationship is linear, modeled by $C = 75h + 50$ . When you encounter cost problems in electrical work, you're typically dealing with linear relationships that combine fixed costs with variable costs. The key is identifying which cost stays constant and which changes with time or usage. Let's break down this pricing structure: the electrician charges a $50 service fee (this is a one-time fixed cost that doesn't change regardless of hours worked) plus $75 per hour (this is the variable cost that increases with each additional hour). To build the equation, start with the variable portion: $75 times the number of hours gives you $75h$ . Then add the fixed service fee to get $C = 75h + 50$ . This matches option B perfectly. Looking at the incorrect choices: Option A reverses the coefficients, suggesting $C = 50h + 75$ , which would mean $50 per hour plus a $75 service fee – the opposite of what's stated. Option C claims the relationship is non-linear with $C = 125h$ , but this eliminates the separate service fee entirely and incorrectly adds the two costs as if they're both hourly rates. Option D suggests $C = 50h^2 + 75$ , making this non-linear with a squared term, which would mean costs accelerate exponentially – unrealistic for standard electrical work pricing. For IBEW exam success, remember that most real-world electrical pricing follows the pattern: Total Cost = Fixed Fee + (Hourly Rate × Hours). Always identify the fixed component first, then add the variable component. Linear relationships in trade work are the norm, so be skeptical of non-linear options unless the situation clearly calls for them.
Which of the following descriptions represents a situation that can be modeled by a linear function?
Answer: A worker's weekly pay which is $20 per hour for up to 40 hours. Linear functions have a constant rate of change, meaning they produce straight lines when graphed. When you encounter problems asking about linear relationships, look for situations where one quantity changes by the same amount each time another quantity increases by one unit. Option A represents a perfect linear function. A worker earning $20 per hour for up to 40 hours creates the relationship: Pay = $20 \times \text{hours}$ . For every additional hour worked, pay increases by exactly $20. This constant rate of change ($20 per hour) makes it linear. Option B describes exponential growth, not linear. When something doubles each hour, you're multiplying by 2 repeatedly (2, 4, 8, 16...), which creates a curve that gets steeper over time. The rate of change keeps increasing rather than staying constant. Option C represents an inverse relationship. As you add more crew members, the time decreases, but not in a straight line. Adding the second worker might cut time in half, but adding the tenth worker won't have the same dramatic effect. This creates a curved relationship. Option D describes exponential decay. Hot components cool rapidly at first, then more slowly as they approach room temperature. The rate of cooling changes continuously, creating a curve that flattens out over time. Remember this pattern: linear functions involve addition or subtraction of constant amounts, while exponential functions involve multiplication or division. On electrical exams, you'll often see linear relationships in pay calculations, material costs per unit, and simple time-distance problems.
Consider the function $f(x) = 8/x$ . Why is this function not linear?
Answer: Because the variable x is in the denominator of a fraction. When you encounter questions about whether a function is linear, you need to recall that a linear function has the form $y = mx + b$ where the variable appears only to the first power and isn't in denominators, under radicals, or in exponents. The function $f(x) = 8/x$ can be rewritten as $f(x) = 8x^{-1}$ , which shows that x has a negative exponent. This violates the fundamental requirement for linearity because the variable x must have an exponent of exactly 1 (first power) in linear functions. Additionally, having x in the denominator creates a reciprocal relationship that produces a hyperbola when graphed, not the straight line that defines linear functions. Looking at the answer choices: Choice A correctly identifies that having the variable in the denominator makes the function non-linear. Choice B is wrong because linear functions don't require a constant term— $y = 2x$ is perfectly linear without adding a constant. Choice C is incorrect because the coefficient being an even integer has no bearing on linearity— $f(x) = 7x$ and $f(x) = 8x$ are both linear regardless of whether the coefficient is odd or even. Choice D is wrong because function notation versus y-notation is simply a matter of how we write the function, not what type of function it is. For IBEW exam success, remember this key principle: a function is linear only when the variable appears to the first power and isn't in denominators, under radicals, or as exponents. If you see $x$ in a denominator, it's automatically non-linear.
Which of the following equations represents a linear function?
Answer: $y = 3x + 5$ When you encounter questions about linear functions on the IBEW exam, you need to identify equations that graph as straight lines. A linear function has the form $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept. The key characteristic is that the variable has an exponent of exactly 1. Looking at option C, $y = 3x + 5$ , this fits the linear function template perfectly. Here, the slope is 3 and the y-intercept is 5. The variable $x$ has an implied exponent of 1, which creates a straight line when graphed. Option A, $y = 4/x$ , can be rewritten as $y = 4x^{-1}$ . Since the exponent is -1 (not 1), this creates a hyperbola, not a straight line. This is a rational function that produces a curved graph with asymptotes. Option B, $y = x^2 - 2$ , is a quadratic function because the variable has an exponent of 2. This creates a parabola that curves upward or downward, definitely not linear. Option D, $y = \sqrt{x} + 1$ , can be written as $y = x^{1/2} + 1$ . The fractional exponent of 1/2 means this is a square root function, which produces a curved graph starting at a point and gradually leveling off. For IBEW math questions, remember this simple rule: linear functions only have variables raised to the first power (exponent of 1), with no variables in denominators, under radicals, or raised to other powers. When in doubt, ask yourself if the graph would be a straight line.
For a function to be linear, the change in the output must be constant for any constant change in the input. Which table below shows a linear function?
Answer: x: -1, 0, 1, 2; y: 1, 4, 7, 10 When you encounter questions about linear functions, remember that linearity means the rate of change (slope) must be constant throughout. For any equal change in x-values, the change in y-values must always be the same. To identify the linear function, calculate the change in y for each unit change in x. Since all tables show x-values increasing by 1 each time, you need consistent y-value changes. Option B shows the linear relationship: from x = -1 to x = 0, y changes from 1 to 4 (change of +3); from x = 0 to x = 1, y changes from 4 to 7 (change of +3); from x = 1 to x = 2, y changes from 7 to 10 (change of +3). The constant change of +3 for each +1 in x confirms this is linear with slope = 3. Option A fails because the y-changes are: +1 to 0 (-1), 0 to 1 (+1), then 1 to 4 (+3) — these aren't constant. Option C shows changes of: -1 to 0 (+1), 0 to 1 (+1), then 1 to 8 (+7) — the dramatic jump to 8 breaks the pattern. Option D has changes of: 2 to 4 (+2), 4 to 6 (+2), then 6 to 9 (+3) — the final change differs from the first two. Study tip: Always check the differences between consecutive y-values when x-values increase by the same amount. If those differences are identical, you've found your linear function. This method works faster than trying to visualize or guess.
A function table is partially shown below. x: 2, 4, 6, 8 y: 5, 10, 15, 20 Which statement correctly describes the function?
Answer: It is a linear function because the rate of change is constant. When you encounter function tables on the IBEW exam, you're being tested on your ability to identify linear relationships by analyzing patterns in the data. The key is to check whether the rate of change between x and y values remains constant. To determine if this function is linear, calculate the rate of change (slope) between consecutive points. From x = 2 to x = 4, y changes from 5 to 10, giving us a rate of change of $\frac{10-5}{4-2} = \frac{5}{2} = 2.5$ . From x = 4 to x = 6, the rate is $\frac{15-10}{6-4} = \frac{5}{2} = 2.5$ . From x = 6 to x = 8, it's $\frac{20-15}{8-6} = \frac{5}{2} = 2.5$ . Since the rate of change is consistently 2.5, this is indeed a linear function. Choice A correctly identifies this constant rate of change as the defining characteristic of linear functions. Choice B incorrectly suggests that having y-values as multiples of 5 prevents linearity—but linear functions can absolutely have outputs that follow patterns like this. Choice C makes the error of focusing on x-values being even numbers, which is irrelevant to linearity; the spacing and relationship between variables matter, not whether they're even or odd. Choice D incorrectly assumes linear functions must pass through the origin—many linear functions have y-intercepts other than zero. For IBEW exam success, remember: linear functions have constant rates of change between any two points. Always calculate the slope between multiple pairs of points to verify consistency.
A relationship between two variables, x and y, is described by the equation $4x - 2y = 8$ . Which statement accurately describes this relationship?
Answer: It is a linear function because it can be rewritten in the form $y = mx + b$ . When you encounter equations on the IBEW exam, you need to identify whether they represent linear or non-linear relationships. A linear function creates a straight line when graphed and can always be written in slope-intercept form: $y = mx + b$ . Let's transform the given equation $4x - 2y = 8$ into slope-intercept form. First, isolate the y-term by subtracting $4x$ from both sides: $-2y = -4x + 8$ . Then divide everything by $-2$ : $y = 2x - 4$ . This is now in the form $y = mx + b$ where the slope $m = 2$ and the y-intercept $b = -4$ . Since we can rewrite it this way, it's definitely a linear function. Answer A is wrong because linear functions graph as straight lines, not parabolas. Parabolas represent quadratic functions that contain squared terms. Answer B incorrectly suggests that having variables on the same side prevents a linear relationship. The position of variables in the original equation doesn't determine linearity—what matters is the highest power of the variables, which is 1 in this case. Answer C confuses the number of terms with linearity. Linear equations can have any number of terms as long as no variable is raised to a power greater than 1. Answer D correctly identifies this as linear because it can be rewritten in slope-intercept form. Remember: On IBEW math problems, always check if you can rearrange an equation into $y = mx + b$ form. If the highest power of any variable is 1, you're dealing with a linear relationship.
Consider the function $f(x) = -x + 10$ . Is this function linear?
Answer: Yes, because it is in the form $y = mx + b$ with a slope of -1. When you encounter questions about linear functions, focus on the fundamental definition: a linear function is any function that can be written in the form $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept. Looking at $f(x) = -x + 10$ , you can rewrite this as $y = -1x + 10$ . This perfectly matches the $y = mx + b$ format with $m = -1$ (the slope) and $b = 10$ (the y-intercept). Since it fits this standard form, it's definitely a linear function. Answer choice A is incorrect because the y-intercept's value doesn't determine whether a function is linear. A linear function can have any y-intercept—positive, negative, or zero. Answer choice B contains a common misconception: the slope's sign is irrelevant to linearity. Linear functions can have positive slopes (increasing), negative slopes (decreasing), or zero slopes (horizontal). Answer choice D describes a property that happens to be true for this particular function when $x > 0$ , but this input-output relationship doesn't define linearity. The correct answer is C because the function is in proper linear form with slope $-1$ . Study tip: Don't get distracted by the specific values of slope or y-intercept when determining linearity. Always check if the function can be written as $y = mx + b$ where both $m$ and $b$ are constants. If yes, it's linear—regardless of whether the slope is positive, negative, or the y-intercept is large or small.
A circuit's voltage V is kept constant. According to Ohm's Law, the relationship between current I and resistance R is $I = V/R$ . If current is considered a function of resistance, is this function linear?
Answer: No, because the independent variable R is in the denominator. This question tests your understanding of linear functions in the context of electrical circuits. When examining whether a relationship is linear, you need to look at the mathematical form, not just whether it involves fundamental principles or positive values. Looking at the equation $I = V/R$ with constant voltage V, you can rewrite this as $I = V \cdot \frac{1}{R}$ or $I = \frac{V}{R}$ . This is an inverse relationship, also called a reciprocal function. In a linear function, the independent variable appears to the first power in the numerator, like $y = mx + b$ . Here, the independent variable R appears in the denominator, creating a hyperbolic curve rather than a straight line. As resistance increases, current decreases at a decreasing rate, which is characteristic of inverse relationships. Answer choice A incorrectly focuses on the sign of the variables. Linearity has nothing to do with whether variables can be negative—it's about the mathematical relationship between them. Choice B makes the error of confusing fundamental electrical principles with mathematical linearity. While Ohm's Law is indeed fundamental, being a basic principle doesn't make a function linear. Choice C incorrectly suggests that having a constant coefficient (V) makes the relationship linear. Even with V constant, the inverse relationship with R still creates a non-linear function. Remember: when determining if a function is linear, look at the mathematical form. If the independent variable appears in the denominator, as an exponent other than 1, or in any other non-additive way, the function isn't linear regardless of the physical principles involved.
Which of the following functions is NOT linear?
Answer: $y = 3x^2 + 1$ When you encounter questions about linear functions, remember that a linear function creates a straight line when graphed and can always be written in the form $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept. The key identifier is that the variable $x$ appears only to the first power (no exponents other than 1). Looking at each option, you can quickly identify which functions follow this linear pattern. Option A, $y = 3x^2 + 1$ , contains $x^2$ , meaning the variable is raised to the second power. This creates a parabola when graphed, making it a quadratic function, not linear. Option B, $y = x/3 + 2$ , can be rewritten as $y = \frac{1}{3}x + 2$ , which fits the $y = mx + b$ form perfectly with slope $\frac{1}{3}$ and y-intercept 2. Option C, $y = 2(x + 4)$ , expands to $y = 2x + 8$ , also fitting the linear form with slope 2 and y-intercept 8. Option D, $y = 5x - 9$ , is already in standard linear form with slope 5 and y-intercept -9. For IBEW exam success, always scan the equation for the highest power of $x$ . If you see $x^2$ , $x^3$ , or any power greater than 1, the function isn't linear. Also watch for variables in denominators (like $\frac{1}{x}$ ) or under square roots, as these also indicate non-linear functions.
A function's values are shown in the table below. x: 0, 1, 2, 3 y: 1, 2, 5, 10 Is the function represented by this table linear?
Answer: No, because the change in y is not constant for a constant change in x. When you encounter a table of values and need to determine if a function is linear, you're testing whether the function has a constant rate of change. A linear function creates a straight line when graphed, which means the change in y-values must be consistent for equal changes in x-values. Let's examine the rate of change between consecutive points. From x = 0 to x = 1, y changes from 1 to 2 (change of +1). From x = 1 to x = 2, y changes from 2 to 5 (change of +3). From x = 2 to x = 3, y changes from 5 to 10 (change of +5). Since the changes in y are +1, +3, and +5 respectively, the rate of change is not constant, making this a nonlinear function. Answer C correctly identifies this. Answer A is wrong because the sign of y-values has nothing to do with linearity—linear functions can have positive, negative, or mixed y-values. Answer B makes a common error by focusing on the x-values, but equal spacing in x-values is necessary but not sufficient for linearity; you must also check if the y-values change at a constant rate. Answer D is incorrect because a linear function can have any y-intercept (the y-value when x = 0)—it doesn't need to start at zero. Remember this key test: for any function to be linear, equal changes in x must produce equal changes in y. Always calculate the differences between consecutive y-values when x-values increase by the same amount.
Identify the equation that represents a linear function.
Answer: $y = x/5 - 1$ Linear functions are fundamental in electrical work, appearing in everything from Ohm's law relationships to load calculations. When you encounter a question about identifying linear functions, you need to recognize the standard form: $y = mx + b$ , where the variable has an exponent of 1 (or no visible exponent). Looking at the correct answer, option C gives us $y = \frac{x}{5} - 1$ . This can be rewritten as $y = \frac{1}{5}x - 1$ , which perfectly matches the linear form $y = mx + b$ where the slope $m = \frac{1}{5}$ and the y-intercept $b = -1$ . The variable $x$ appears to the first power only. Now let's see why the other options fail. Option A, $y = x^5 - 1$ , is a polynomial function where $x$ is raised to the fifth power, making it non-linear. Option B, $y = \frac{5}{x} - 1$ , can be written as $y = 5x^{-1} - 1$ , which is a rational function since $x$ appears in the denominator with a negative exponent. Option D, $y = 5^x - 1$ , is an exponential function where the variable $x$ appears as an exponent rather than as the base. Remember this key pattern: linear functions have the variable raised only to the first power, with no variables in denominators or exponents. On the IBEW exam, you'll use linear relationships constantly in electrical calculations, so mastering this identification skill early will serve you well throughout your career.
Which of the following sets of ordered pairs could NOT be part of a linear function?
Answer: $(1, 1), (2, 4), (3, 9)$ When you encounter ordered pairs and need to determine if they form a linear function, you're testing whether the points create a straight line with a constant rate of change (slope). A linear function has the same slope between any two points. To check this, calculate the slope between consecutive pairs using $\frac{y_2 - y_1}{x_2 - x_1}$ . Let's examine option A: $(1, 1), (2, 4), (3, 9)$ . The slope from the first to second point is $\frac{4-1}{2-1} = 3$ . The slope from the second to third point is $\frac{9-4}{3-2} = 5$ . Since the slopes are different (3 ≠ 5), these points cannot form a linear function. This is your answer. Option B: $(1, 2), (2, 4), (3, 6)$ has consistent slopes of $\frac{4-2}{2-1} = 2$ and $\frac{6-4}{3-2} = 2$ . This represents the linear function $y = 2x$ . Option C: $(0, 5), (1, 5), (2, 5)$ has slopes of $\frac{5-5}{1-0} = 0$ and $\frac{5-5}{2-1} = 0$ . This is a horizontal line, $y = 5$ , which is linear. Option D: $(2, 3), (4, 7), (5, 9)$ has slopes of $\frac{7-3}{4-2} = 2$ and $\frac{9-7}{5-4} = 2$ . This represents $y = 2x - 1$ . Study tip: Always check slope consistency between all consecutive pairs. If any two slopes differ, the relationship isn't linear. Watch for patterns like perfect squares (as in option A) - they often indicate quadratic, not linear, relationships.
Consider the following set of ordered pairs: $(1, 5), (3, 9), (5, 13), (7, 17)$ . Does this set of points represent a linear function?
Answer: Yes, because for every 2-unit increase in x, y consistently increases by 4. When you encounter ordered pairs and need to determine if they represent a linear function, you're testing whether there's a constant rate of change between the variables. A linear function maintains the same slope throughout all points. To check if these points form a linear function, calculate the rate of change (slope) between consecutive pairs. From (1, 5) to (3, 9): the change in x is 2 and the change in y is 4, giving a slope of $\frac{4}{2} = 2$ . From (3, 9) to (5, 13): again, x increases by 2 and y increases by 4, maintaining the slope of 2. The same pattern continues from (5, 13) to (7, 17). Since the slope remains constant at 2, this is indeed a linear function with the equation $y = 2x + 3$ . Choice A incorrectly focuses on whether x-coordinates are odd numbers, which is irrelevant to linearity. Choice B makes the false assumption that y-values must be consecutive integers for a linear relationship—this completely misunderstands what makes a function linear. Choice D incorrectly suggests that the starting values determine linearity, when it's actually the consistent rate of change that matters. For IBEW electrical calculations, remember that linear relationships appear frequently in circuits (like Ohm's law applications). Always look for constant rates of change when identifying linear functions—check if equal changes in the input consistently produce equal changes in the output.
A job requires digging a trench. The time (t) in hours to complete the job depends on the number of workers (w) and is modeled by $t = 24/w$ . This relationship is:
Answer: Non-linear, because the independent variable w is in the denominator. When you encounter equations in mathematical modeling, the key distinction between linear and non-linear relationships lies in how the variables are arranged, not in the real-world behavior they describe. A linear relationship has the form $y = mx + b$ , where variables appear only to the first power and are not multiplied together, divided by each other, or appear in denominators. In contrast, a non-linear relationship involves variables raised to powers other than 1, multiplied together, or appearing in denominators. The equation $t = 24/w$ can be rewritten as $t = 24w^{-1}$ , showing that the variable $w$ has an exponent of -1. This immediately identifies it as non-linear because the independent variable appears in the denominator (or equivalently, has a negative exponent). Choice A correctly identifies this as non-linear specifically because $w$ is in the denominator, which is the defining characteristic that makes this relationship non-linear. Choice B confuses the direction of change with linearity. While it's true that time decreases as workers increase, this describes an inverse relationship, not linearity. The mathematical form, not the direction of change, determines linearity. Choice C incorrectly focuses on the practical constraint that worker count can't be negative. Domain restrictions don't determine whether a function is linear or non-linear. Choice D mistakenly suggests that real-world scenarios automatically create linear relationships, which is mathematically irrelevant. Remember: linearity depends on mathematical form, not real-world behavior. Look for variables in denominators, raised to powers other than 1, or multiplied together—these signal non-linear relationships.
Ohm's Law is linear; which function is also linear and would graph as a straight line?
Answer: $y=-7x+3$ This question tests understanding of linear functions in algebra, specifically identifying linear behavior in equations and graphs. Linear functions are characterized by a constant rate of change and can be graphically represented by a straight line. In this question, each option is presented as an equation, testing your ability to discern linearity. The correct choice is A because it represents a function following the form y = mx + b, indicating linearity. A common distractor is C, which appears linear but involves an exponent, a common misconception when interpreting linear functions. To master this concept, practice identifying the slope and y-intercept in equations and distinguish between linear and non-linear graphs. Encourage recognition of linear forms in real-world contexts, such as direct proportionality in electrical circuits.
Which statement below correctly describes a defining property of all linear functions?
Answer: The graph of the function is a non-vertical straight line. When you encounter questions about linear functions, focus on their most fundamental characteristic: they create straight-line graphs when plotted on a coordinate plane. A linear function has the general form $f(x) = mx + b$ , where $m$ is the slope and $b$ is the y-intercept. The defining property of all linear functions is that their rate of change (slope) remains constant between any two points. This constant rate of change is what creates the straight-line graph, making answer D correct. Let's examine why the other options are incorrect: Answer A is wrong because linear functions frequently include fractions and decimals. For example, $f(x) = 0.5x + 2.75$ is perfectly linear despite containing decimals. Answer B represents a common misconception. While some linear functions pass through the origin (like $f(x) = 3x$ ), many don't. The function $f(x) = 2x + 5$ is linear but passes through (0, 5), not the origin. Only linear functions where $b = 0$ pass through the origin. Answer C is false because linear functions can have any relationship between input and output values. With a negative slope like $f(x) = -2x + 1$ , the output decreases as input increases, and there are many points where the output is less than the input. Remember this key distinction: linear functions always produce straight-line graphs (non-vertical, since vertical lines aren't functions). This visual characteristic is the most reliable way to identify linear relationships in electrical applications like voltage-current relationships in resistive circuits.
A sequence of numbers is generated by the rule "start with 2, then add 5 to the previous number to get the next." If x is the term number (1st, 2nd, 3rd...) and y is the value of the term, is the function that relates x and y linear?
Answer: Yes, because there is a constant rate of change (+5) for each 1-unit step in x. When you encounter questions about sequences and whether they represent linear functions, focus on the key characteristic of linear functions: a constant rate of change. This means that for every one-unit increase in the input (x), the output (y) changes by the same amount. Let's examine this sequence: starting with 2, then adding 5 each time gives us 2, 7, 12, 17, 22... To determine if this is linear, look at how y changes as x increases by 1. From the 1st term to 2nd term: y increases by 5. From 2nd to 3rd: y increases by 5. This pattern continues—the rate of change is constantly +5, which is the hallmark of a linear function. Choice D correctly identifies this constant rate of change as the reason the function is linear. The relationship can be expressed as $y = 5x - 3$ , which is in the standard linear form $y = mx + b$ . Choice A is wrong because being positive integers has nothing to do with linearity—linear functions can include negative numbers, fractions, or decimals. Choice B incorrectly focuses on the starting value; linear functions don't need to start at zero (that would just mean the y-intercept is zero). Choice C misunderstands the role of addition versus multiplication—linear functions are defined by having a constant rate of change, regardless of whether the rule uses addition or multiplication. Remember: For IBEW math questions involving sequences, always check if there's a constant difference between consecutive terms. If yes, it's linear.

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