IBEW: Electrical Training Alliance Aptitude Test Quiz: Identify Linear Functions
20 questions · exam conditions
0:00
Identify Linear FunctionsQuestion 1 of 20

Ohm's Law is linear; which function is also linear and would graph as a straight line?

y=−7x+3y=-7x+3
y=7x+3y=\dfrac{7}{x}+3
y=x2+3y=x^2+3
y=∣x∣+3y=|x|+3
← Back to quizzes

IBEW: Electrical Training Alliance Aptitude Test Quiz

IBEW: Electrical Training Alliance Aptitude Test Quiz: Identify Linear Functions

Practice Identify Linear Functions in IBEW: Electrical Training Alliance Aptitude Test with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Identify Linear Functions, giving you a quick way to practice the rules, question types, and explanations that matter most for IBEW: Electrical Training Alliance Aptitude Test.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Ohm's Law is linear; which function is also linear and would graph as a straight line?

  1. y=−7x+3y=-7x+3 (correct answer)
  2. y=7x+3y=\dfrac{7}{x}+3
  3. y=x2+3y=x^2+3
  4. y=∣x∣+3y=|x|+3
Explanation: 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.

Question 2

Which of the following sets of ordered pairs could NOT be part of a linear function?

  1. (1,1),(2,4),(3,9)(1, 1), (2, 4), (3, 9) (correct answer)
  2. (1,2),(2,4),(3,6)(1, 2), (2, 4), (3, 6)
  3. (0,5),(1,5),(2,5)(0, 5), (1, 5), (2, 5)
  4. (2,3),(4,7),(5,9)(2, 3), (4, 7), (5, 9)
Explanation: 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 y2−y1x2−x1\frac{y_2 - y_1}{x_2 - x_1}. Let's examine option A: (1,1),(2,4),(3,9)(1, 1), (2, 4), (3, 9). The slope from the first to second point is 4−12−1=3\frac{4-1}{2-1} = 3. The slope from the second to third point is 9−43−2=5\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)(1, 2), (2, 4), (3, 6) has consistent slopes of 4−22−1=2\frac{4-2}{2-1} = 2 and 6−43−2=2\frac{6-4}{3-2} = 2. This represents the linear function y=2xy = 2x. Option C: (0,5),(1,5),(2,5)(0, 5), (1, 5), (2, 5) has slopes of 5−51−0=0\frac{5-5}{1-0} = 0 and 5−52−1=0\frac{5-5}{2-1} = 0. This is a horizontal line, y=5y = 5, which is linear. Option D: (2,3),(4,7),(5,9)(2, 3), (4, 7), (5, 9) has slopes of 7−34−2=2\frac{7-3}{4-2} = 2 and 9−75−4=2\frac{9-7}{5-4} = 2. This represents y=2x−1y = 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.

Question 3

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?

  1. Yes, because the numbers in the sequence are all positive integers.
  2. No, because the first term in the sequence is not zero.
  3. No, because the rule only involves addition, not multiplication.
  4. Yes, because there is a constant rate of change (+5) for each 1-unit step in x. (correct answer)
Explanation: 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−3y = 5x - 3, which is in the standard linear form y=mx+by = 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.

Question 4

A circuit's voltage V is kept constant. According to Ohm's Law, the relationship between current I and resistance R is I=V/RI = V/R. If current is considered a function of resistance, is this function linear?

  1. No, because current and resistance cannot have negative values.
  2. Yes, because Ohm's Law is a fundamental principle in electricity.
  3. Yes, because the voltage V is a constant value in this case.
  4. No, because the independent variable R is in the denominator. (correct answer)
Explanation: 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/RI = V/R with constant voltage V, you can rewrite this as I=V⋅1RI = V \cdot \frac{1}{R} or I=VRI = \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+by = 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.

Question 5

The power P (in watts) in a circuit is related to the current I (in amps) and a constant resistance R by the formula P=I2RP = I^2R. If power is considered a function of current, is this relationship linear?

  1. No, because the variable for current, I, is squared. (correct answer)
  2. Yes, because the resistance R is a constant value.
  3. Yes, because power and current are directly related in circuits.
  4. No, because the formula does not contain an addition or subtraction.
Explanation: When you encounter questions about mathematical relationships in electrical circuits, focus on the fundamental definition of linearity: a linear relationship means one variable changes at a constant rate relative to another, creating a straight line when graphed. Looking at the power formula P=I2RP = I^2R, you need to examine how power changes as current changes. Since current is squared in this equation, doubling the current doesn't just double the power—it quadruples it. Triple the current, and power increases nine times. This creates a curved (parabolic) relationship, not a straight line, making it nonlinear. Choice A is correct because the squared term I2I^2 creates this nonlinear relationship. The exponent of 2 means the rate of change in power accelerates as current increases. Choice B incorrectly assumes that having a constant (R) makes the relationship linear. While R being constant is important, it doesn't change the fact that I is still squared. Even P=5I2P = 5I^2 (if R = 5) would still be nonlinear. Choice C confuses "directly related" with "linear." Yes, as current increases, power increases, so they're directly related. However, direct relationship doesn't mean linear—the rate of increase matters. Choice D misunderstands what makes relationships linear or nonlinear. Addition and subtraction don't determine linearity; it's about whether variables have exponents other than 1. P=2I+5P = 2I + 5 would be linear, while P=I2P = I^2 is not. Remember: On IBEW exams, when analyzing mathematical relationships, always check the exponents first. Any variable raised to a power other than 1 indicates a nonlinear relationship.

Question 6

Consider the following set of ordered pairs: (1,5),(3,9),(5,13),(7,17)(1, 5), (3, 9), (5, 13), (7, 17). Does this set of points represent a linear function?

  1. No, because the relationship involves an odd number for the x-coordinate.
  2. No, because the y-values are not consecutive integers like the x-values.
  3. Yes, because for every 2-unit increase in x, y consistently increases by 4. (correct answer)
  4. Yes, because the first x-value is 1 and the first y-value is a multiple of 1.
Explanation: 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 42=2\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+3y = 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.

Question 7

A function table is partially shown below. x: 2, 4, 6, 8 y: 5, 10, 15, 20 Which statement correctly describes the function?

  1. It is a linear function because the rate of change is constant. (correct answer)
  2. It is not a linear function because the y-values are multiples of 5.
  3. It is a linear function because all the x-values are even numbers.
  4. It is not a linear function because the point (0,0) is not included.
Explanation: 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 10−54−2=52=2.5\frac{10-5}{4-2} = \frac{5}{2} = 2.5. From x = 4 to x = 6, the rate is 15−106−4=52=2.5\frac{15-10}{6-4} = \frac{5}{2} = 2.5. From x = 6 to x = 8, it's 20−158−6=52=2.5\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.

Question 8

The relationship between the side length (s) of a square and its area (A) is given by the formula A=s2A = s^2. This relationship is:

  1. Linear, because the formula does not contain a y-intercept.
  2. Linear, because for every increase in s, A also increases.
  3. Non-linear, because the variable s is squared. (correct answer)
  4. Non-linear, because area cannot be represented by an equation.
Explanation: When you encounter questions about mathematical relationships, you need to distinguish between linear and non-linear functions by examining how the variables relate to each other, not just whether the function increases or decreases. A linear relationship means that when you plot the function on a graph, you get a straight line. This happens when the variable appears to the first power only. The general form is y=mx+by = mx + b, where the highest exponent on any variable is 1. In contrast, a non-linear relationship creates a curved line when graphed, typically because variables are raised to powers other than 1. In the formula A=s2A = s^2, the variable ss is raised to the second power (squared), which immediately tells you this is a non-linear relationship. When you graph this function, you get a parabola, not a straight line. This makes option C correct. Option A is wrong because the presence or absence of a y-intercept doesn't determine whether a relationship is linear—many linear functions pass through the origin with no y-intercept. Option B makes a common mistake by confusing "always increasing" with "linear." While it's true that as ss increases, AA increases, this doesn't make the relationship linear. A function can increase while still being non-linear. Option D is simply false—area relationships can definitely be represented by equations. Study tip: Look at the exponents first. If any variable has an exponent other than 1, the relationship is non-linear, regardless of whether the function increases, decreases, or has intercepts.

Question 9

The number of feet (F) in a given number of inches (I) is represented by the formula F=I/12F = I/12. This relationship is:

  1. Non-linear, because the result is often a fraction or decimal.
  2. Non-linear, because the formula involves a division operation.
  3. Linear, as it can be written in the form F=(1/12)I+0F = (1/12)I + 0. (correct answer)
  4. Linear, but only for values of I that are multiples of 12.
Explanation: When you encounter questions about mathematical relationships, you need to identify whether the relationship between variables is linear or non-linear. A linear relationship means the graph forms a straight line and can be written in the form y=mx+by = mx + b, where mm is the slope and bb is the y-intercept. The formula F=I/12F = I/12 can be rewritten as F=112I+0F = \frac{1}{12}I + 0. This perfectly matches the linear form y=mx+by = mx + b, where the slope is 112\frac{1}{12} and the y-intercept is 0. Since the relationship maintains a constant rate of change (every 12 inches always equals exactly 1 foot), it's definitively linear. Answer C correctly identifies this linear relationship. Answer A incorrectly suggests that getting fractional or decimal results makes a relationship non-linear. The nature of the output (whole numbers versus fractions) has nothing to do with linearity – it's about the mathematical relationship between variables. Answer B makes the common mistake of thinking division operations automatically create non-linear relationships. However, division by a constant (like 12) maintains linearity. Non-linearity involves operations like squaring variables, taking roots, or exponentials. Answer D wrongly claims the relationship is only linear for multiples of 12. Linear relationships maintain their properties for all valid input values, not just convenient whole number outputs. Remember: linearity depends on the mathematical form of the equation, not the type of numbers it produces. Look for the y=mx+by = mx + b pattern to identify linear relationships quickly.

Question 10

Which of the following equations represents a linear function?

  1. y=2xy = 2^x
  2. y=∣x∣+2y = |x| + 2
  3. y−3x=7y - 3x = 7 (correct answer)
  4. xy=10xy = 10
Explanation: When you encounter equations on the IBEW exam, you need to recognize the difference between linear and nonlinear functions. A linear function creates a straight line when graphed and can always be written in the form y=mx+by = mx + b or Ax+By=CAx + By = C, where the variables have exponents of 1. Option C, y−3x=7y - 3x = 7, is the linear function here. You can rearrange it to standard form: y=3x+7y = 3x + 7. This has a constant rate of change (slope = 3) and will graph as a straight line. Both variables have exponents of 1, which is the defining characteristic of linear functions. Now let's examine why the other options are nonlinear. Option A, y=2xy = 2^x, is an exponential function because the variable xx appears in the exponent, creating a curved graph that increases rapidly. Option B, y=∣x∣+2y = |x| + 2, is an absolute value function that creates a V-shaped graph due to the absolute value operation changing the behavior of negative xx values. Option D, xy=10xy = 10, is a rational function (hyperbola) because it can be rewritten as y=10xy = \frac{10}{x}, where xx appears in the denominator with a negative exponent. For IBEW exam success, remember this key identifier: linear functions have variables with exponents of exactly 1, no variables in denominators, and no special operations like absolute value or exponentials. When in doubt, try to rearrange the equation into y=mx+by = mx + b form.

Question 11

Consider the function f(x)=−x+10f(x) = -x + 10. Is this function linear?

  1. No, because the y-intercept is a positive number greater than one.
  2. No, because the slope of the function is a negative number.
  3. Yes, because it is in the form y=mx+by = mx + b with a slope of -1. (correct answer)
  4. Yes, because for any positive x, the output is smaller than the input.
Explanation: 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+by = mx + b, where mm is the slope and bb is the y-intercept. Looking at f(x)=−x+10f(x) = -x + 10, you can rewrite this as y=−1x+10y = -1x + 10. This perfectly matches the y=mx+by = mx + b format with m=−1m = -1 (the slope) and b=10b = 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>0x > 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 −1-1. 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+by = mx + b where both mm and bb are constants. If yes, it's linear—regardless of whether the slope is positive, negative, or the y-intercept is large or small.

Question 12

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?

  1. x: -1, 0, 1, 2; y: 1, 0, 1, 4
  2. x: -1, 0, 1, 2; y: 1, 4, 7, 10 (correct answer)
  3. x: -1, 0, 1, 2; y: -1, 0, 1, 8
  4. x: -1, 0, 1, 2; y: 2, 4, 6, 9
Explanation: 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.

Question 13

Which of the following functions exhibits a non-constant rate of change and is therefore non-linear?

  1. f(x)=x2+xf(x) = x^2 + x (correct answer)
  2. f(x)=100−5xf(x) = 100 - 5x
  3. f(x)=(x/2)f(x) = (x/2)
  4. f(x)=3(x−1)f(x) = 3(x-1)
Explanation: When you encounter questions about linear versus non-linear functions, focus on the rate of change. Linear functions have a constant rate of change (the same slope everywhere), while non-linear functions have rates of change that vary depending on where you are on the curve. To identify which function is non-linear, look for the highest power of x in each equation. Linear functions contain only x to the first power, while non-linear functions contain x raised to powers other than 1. Choice A, f(x)=x2+xf(x) = x^2 + x, contains an x2x^2 term, making it a quadratic function. The rate of change varies because as x increases, the x2x^2 term grows much faster than the linear xx term. This creates a curve rather than a straight line, confirming it's non-linear. Choice B, f(x)=100−5xf(x) = 100 - 5x, is linear because it can be written as f(x)=−5x+100f(x) = -5x + 100, following the form y=mx+by = mx + b with a constant slope of -5. Choice C, f(x)=x/2f(x) = x/2, is linear because it's equivalent to f(x)=0.5xf(x) = 0.5x, which has a constant slope of 0.5. Choice D, f(x)=3(x−1)f(x) = 3(x-1), expands to f(x)=3x−3f(x) = 3x - 3, which is linear with a constant slope of 3. For IBEW exam success, remember this quick check: if the highest power of x is anything other than 1 (like x2x^2, x3x^3, or x\sqrt{x}), the function is non-linear. Only first-degree polynomials produce straight lines with constant rates of change.

Question 14

The equation x=7x = 7 represents a vertical line. Is this a linear function?

  1. No, because it is a vertical line and fails the vertical line test for functions. (correct answer)
  2. Yes, because its graph is a straight line and is therefore linear.
  3. Yes, because it can be written in the standard form Ax+By=CAx + By = C.
  4. No, because the equation does not contain a y variable.
Explanation: When you encounter equations like x=7x = 7 on the IBEW exam, you need to distinguish between linear equations and linear functions—they're not the same thing. This question tests your understanding of what qualifies as a function. The equation x=7x = 7 does represent a vertical line, but it's not a function at all. Here's why: for any function, each input (x-value) must correspond to exactly one output (y-value). However, the vertical line x=7x = 7 assigns the same x-value (7) to infinitely many different y-values. If you pick any y-value—say 2, 10, or -5—the point (7, 2), (7, 10), and (7, -5) are all on this line. This violates the definition of a function, which is precisely what the vertical line test checks for. Looking at the wrong answers: Choice B incorrectly assumes that all straight lines represent linear functions—this is a common misconception. Choice C confuses linear equations with linear functions; while x=7x = 7 can be written as 1x+0y=71x + 0y = 7, fitting the form Ax+By=CAx + By = C, this doesn't make it a function. Choice D identifies a true fact (no y-variable) but misses the fundamental issue—the absence of y doesn't disqualify something from being a function, but the failure of the vertical line test does. Remember this key distinction: all linear functions produce straight lines, but not all straight lines represent linear functions. Vertical lines always fail to be functions because they violate the "one output per input" rule.

Question 15

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?

  1. Yes, because all the y-values are positive numbers.
  2. Yes, because the x-values are increasing by a constant amount of 1.
  3. No, because the change in y is not constant for a constant change in x. (correct answer)
  4. No, because the first y-value in the table is not zero.
Explanation: 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.

Question 16

A function is defined by the equation y=5y = 5. Is this a linear function?

  1. Yes, but only if the value of x is also equal to 5.
  2. No, because there is no x variable present in the equation.
  3. No, because the output value of the function never changes.
  4. Yes, it represents a horizontal line with a slope of zero. (correct answer)
Explanation: 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+by = mx + b, where mm is the slope and bb is the y-intercept. The key insight is that this includes cases where the slope equals zero. The equation y=5y = 5 is indeed a linear function because it can be rewritten as y=0x+5y = 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=5y = 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=5y = 5 implicitly has an x term with coefficient zero (y=0x+5y = 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=ky = 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.

Question 17

Which statement below correctly describes a defining property of all linear functions?

  1. The function does not include any fractions or decimals.
  2. The graph of the function must pass through the origin (0,0).
  3. The output value is always greater than the input value.
  4. The graph of the function is a non-vertical straight line. (correct answer)
Explanation: 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+bf(x) = mx + b, where mm is the slope and bb 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+2.75f(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)=3xf(x) = 3x), many don't. The function f(x)=2x+5f(x) = 2x + 5 is linear but passes through (0, 5), not the origin. Only linear functions where b=0b = 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+1f(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.

Question 18

A function's relationship is stated as "the output y is the cube of the input x". Which equation represents this, and is it linear?

  1. y=3xy = 3x; it is linear.
  2. y=x3y = x^3; it is not linear. (correct answer)
  3. y=x+3y = x+3; it is linear.
  4. y=x3y = x^3; it is linear.
Explanation: When you encounter function relationship problems, focus on translating the verbal description into mathematical notation and then determining if the relationship follows a straight line pattern. The phrase "y is the cube of x" means you take the input value x and raise it to the third power to get the output y. This translates directly to y=x3y = x^3. To determine if a function is linear, ask whether it graphs as a straight line or, algebraically, whether it can be written in the form y=mx+by = mx + b where m and b are constants. The function y=x3y = x^3 is not linear because it contains a variable raised to a power other than 1. When you graph y=x3y = x^3, you get a curved S-shape that passes through the origin, not a straight line. This makes option B correct. Option A gives y=3xy = 3x, which means "y equals three times x," not "y equals x cubed." While this is linear (it graphs as a straight line), it doesn't match the given relationship. Option C shows y=x+3y = x + 3, meaning "y equals x plus three," which is both the wrong relationship and represents a different linear function. Option D correctly identifies the equation as y=x3y = x^3 but incorrectly claims it's linear. Remember this key distinction: linear functions have variables raised only to the first power and graph as straight lines, while polynomial functions with higher powers (like cubes) create curved graphs and are nonlinear.

Question 19

Which of the following scenarios describes a relationship that is a linear function?

  1. The total cost of buying spools of wire at $45 per spool. (correct answer)
  2. The area of a circular workspace as a function of its radius.
  3. The height of a ball that is thrown into the air as a function of time.
  4. The remaining amount of a substance that halves every year.
Explanation: 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+by = 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 $Cost=45×Number of spools\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.

Question 20

Which of the following equations represents a linear function?

  1. y=4/xy = 4/x
  2. y=x2−2y = x^2 - 2
  3. y=3x+5y = 3x + 5 (correct answer)
  4. y=x+1y = \sqrt{x} + 1
Explanation: 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+by = mx + b, where mm is the slope and bb is the y-intercept. The key characteristic is that the variable has an exponent of exactly 1. Looking at option C, y=3x+5y = 3x + 5, this fits the linear function template perfectly. Here, the slope is 3 and the y-intercept is 5. The variable xx has an implied exponent of 1, which creates a straight line when graphed. Option A, y=4/xy = 4/x, can be rewritten as y=4x−1y = 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=x2−2y = 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=x+1y = \sqrt{x} + 1, can be written as y=x1/2+1y = 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.