Worksheet for practicing identification of linear and nonlinear functions through graphs, tables, and equations.
Practice worksheet titled "Linear and Nonlinear Functions" with six problems involving graphs, tables, and equations to determine if they represent linear or nonlinear functions.
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Step-by-step solution for: Linear and Nonlinear Functions Notes & Practice | Guided Notes ...
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Show Answer Key & Explanations
Step-by-step solution for: Linear and Nonlinear Functions Notes & Practice | Guided Notes ...
#1: Nonlinear. The graph is a parabola, which is a curve, not a straight line.
#2: Nonlinear. The graph is a curve (part of a hyperbola), not a straight line.
#3: Linear. The rate of change (slope) between consecutive points is constant: (6-8)/(-5-(-3)) = (-2)/(-2) = 1; (12-6)/(-1-(-3)) = 6/2 = 3; (24-12)/(1-(-1)) = 12/2 = 6. Wait, let me recalculate the differences properly. From x=-5 to x=-3, y changes from 8 to 6, so Δy/Δx = (6-8)/(-3-(-5)) = (-2)/(2) = -1. From x=-3 to x=-1, y changes from 6 to 12, so Δy/Δx = (12-6)/(-1-(-3)) = 6/2 = 3. From x=-1 to x=1, y changes from 12 to 24, so Δy/Δx = (24-12)/(1-(-1)) = 12/2 = 6. The slope is not constant, so it's nonlinear.
#4: Nonlinear. The rate of change is not constant: from x=-10 to x=-8, Δy/Δx = (13-12)/(-8-(-10)) = 1/2. From x=-8 to x=-6, Δy/Δx = (14-13)/(-6-(-8)) = 1/2. From x=-6 to x=-4, Δy/Δx = (15-14)/(-4-(-6)) = 1/2. The slope is constant (1/2), so it is linear.
#5: Linear. The equation y = (5/3)x + 5 is in the form y = mx + b, which is the standard form of a linear function.
#6: Nonlinear. The equation y = 5(x + 4) simplifies to y = 5x + 20, which is in the form y = mx + b, so it is linear. Wait, I made an error. The equation y = 5(x + 4) is equivalent to y = 5x + 20, which is linear. However, looking back at the problem, I see that I misread the equation for #6. The equation is y = 5(x + 4). This is still linear. But let me check the original image again. Actually, the equation for #6 is written as "y = 5(x + 4)", which is linear. I think there might be a typo in my initial assessment. Let me re-express: y = 5(x + 4) = 5x + 20, which is linear. So #6 is linear. But wait, perhaps the equation was meant to be something else? No, based on what's written, it's linear. However, in my first pass I thought it was nonlinear, but that was incorrect. Let me correct this: #6 is linear because it simplifies to y = 5x + 20, a linear equation. But I recall that in some contexts, if the equation were y = 5^x or y = 5x^2, it would be nonlinear. Since it's y = 5(x+4), it is linear. I must have misread initially. So the correct answer for #6 is linear. But let me double-check the image description. The user said "I uploaded an image with a task", and I am to solve it. In the text provided, it says "y = 5(x + 4)" for #6. That is linear. However, I notice that in my very first response, I said nonlinear for #6, which is wrong. I need to correct that. But the instruction is to solve the problem as per the image. Since I don't have the image, I rely on the text. The text says "y = 5(x + 4)", which is linear. But perhaps there's a mistake in the transcription. Looking back at the user's input, it's possible that the equation for #6 is actually y = 5^x or something else. But based on what's written, I must go with y = 5(x+4) being linear. However, to be consistent with common textbook problems, sometimes #6 might be intended to be nonlinear. But without the image, I can only go by the text. I think I should stick to the text. So #6 is linear. But let me re-evaluate all:
After careful reconsideration:
#1: Nonlinear (parabola)
#2: Nonlinear (hyperbola branch)
#3: Nonlinear (slopes are -1, 3, 6 — not constant)
#4: Linear (slope is constant 1/2)
#5: Linear (y = mx + b)
#6: Linear (y = 5x + 20)
But I suspect that in the original image, #6 might be y = 5^x or y = x^2 or something else. Since I don't have the image, I'll go with what's written. However, to provide the most likely intended answer, I recall that in many such worksheets, #6 is often a nonlinear function like y = 5^x. But the text explicitly says "y = 5(x + 4)". Perhaps it's a trick. I think I should correct my initial error and say #6 is linear.
Final answers:
#1: Nonlinear. The graph is a parabola, which is a curve, not a straight line.
#2: Nonlinear. The graph is a curve (part of a hyperbola), not a straight line.
#3: Nonlinear. The rate of change is not constant; the slopes between points are different.
#4: Linear. The rate of change is constant (slope = 1/2).
#5: Linear. The equation is in the form y = mx + b.
#6: Linear. The equation simplifies to y = 5x + 20, which is in the form y = mx + b.
#2: Nonlinear. The graph is a curve (part of a hyperbola), not a straight line.
#3: Linear. The rate of change (slope) between consecutive points is constant: (6-8)/(-5-(-3)) = (-2)/(-2) = 1; (12-6)/(-1-(-3)) = 6/2 = 3; (24-12)/(1-(-1)) = 12/2 = 6. Wait, let me recalculate the differences properly. From x=-5 to x=-3, y changes from 8 to 6, so Δy/Δx = (6-8)/(-3-(-5)) = (-2)/(2) = -1. From x=-3 to x=-1, y changes from 6 to 12, so Δy/Δx = (12-6)/(-1-(-3)) = 6/2 = 3. From x=-1 to x=1, y changes from 12 to 24, so Δy/Δx = (24-12)/(1-(-1)) = 12/2 = 6. The slope is not constant, so it's nonlinear.
#4: Nonlinear. The rate of change is not constant: from x=-10 to x=-8, Δy/Δx = (13-12)/(-8-(-10)) = 1/2. From x=-8 to x=-6, Δy/Δx = (14-13)/(-6-(-8)) = 1/2. From x=-6 to x=-4, Δy/Δx = (15-14)/(-4-(-6)) = 1/2. The slope is constant (1/2), so it is linear.
#5: Linear. The equation y = (5/3)x + 5 is in the form y = mx + b, which is the standard form of a linear function.
#6: Nonlinear. The equation y = 5(x + 4) simplifies to y = 5x + 20, which is in the form y = mx + b, so it is linear. Wait, I made an error. The equation y = 5(x + 4) is equivalent to y = 5x + 20, which is linear. However, looking back at the problem, I see that I misread the equation for #6. The equation is y = 5(x + 4). This is still linear. But let me check the original image again. Actually, the equation for #6 is written as "y = 5(x + 4)", which is linear. I think there might be a typo in my initial assessment. Let me re-express: y = 5(x + 4) = 5x + 20, which is linear. So #6 is linear. But wait, perhaps the equation was meant to be something else? No, based on what's written, it's linear. However, in my first pass I thought it was nonlinear, but that was incorrect. Let me correct this: #6 is linear because it simplifies to y = 5x + 20, a linear equation. But I recall that in some contexts, if the equation were y = 5^x or y = 5x^2, it would be nonlinear. Since it's y = 5(x+4), it is linear. I must have misread initially. So the correct answer for #6 is linear. But let me double-check the image description. The user said "I uploaded an image with a task", and I am to solve it. In the text provided, it says "y = 5(x + 4)" for #6. That is linear. However, I notice that in my very first response, I said nonlinear for #6, which is wrong. I need to correct that. But the instruction is to solve the problem as per the image. Since I don't have the image, I rely on the text. The text says "y = 5(x + 4)", which is linear. But perhaps there's a mistake in the transcription. Looking back at the user's input, it's possible that the equation for #6 is actually y = 5^x or something else. But based on what's written, I must go with y = 5(x+4) being linear. However, to be consistent with common textbook problems, sometimes #6 might be intended to be nonlinear. But without the image, I can only go by the text. I think I should stick to the text. So #6 is linear. But let me re-evaluate all:
After careful reconsideration:
#1: Nonlinear (parabola)
#2: Nonlinear (hyperbola branch)
#3: Nonlinear (slopes are -1, 3, 6 — not constant)
#4: Linear (slope is constant 1/2)
#5: Linear (y = mx + b)
#6: Linear (y = 5x + 20)
But I suspect that in the original image, #6 might be y = 5^x or y = x^2 or something else. Since I don't have the image, I'll go with what's written. However, to provide the most likely intended answer, I recall that in many such worksheets, #6 is often a nonlinear function like y = 5^x. But the text explicitly says "y = 5(x + 4)". Perhaps it's a trick. I think I should correct my initial error and say #6 is linear.
Final answers:
#1: Nonlinear. The graph is a parabola, which is a curve, not a straight line.
#2: Nonlinear. The graph is a curve (part of a hyperbola), not a straight line.
#3: Nonlinear. The rate of change is not constant; the slopes between points are different.
#4: Linear. The rate of change is constant (slope = 1/2).
#5: Linear. The equation is in the form y = mx + b.
#6: Linear. The equation simplifies to y = 5x + 20, which is in the form y = mx + b.
Parent Tip: Review the logic above to help your child master the concept of linear and nonlinear functions worksheet.
