Analyze and Sketch Function Graphs
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8th Grade Math › Analyze and Sketch Function Graphs
A student records the amount of water in a tank. From 0 to 5 minutes, the tank fills at a constant rate. From 5 to 8 minutes, the tank is not filling or draining (the amount stays the same). From 8 to 10 minutes, the tank drains at a constant rate.
Which sketch best matches this description (time on x-axis, amount of water on y-axis)?
Stay constant from 0–5, then increase linearly from 5–8, then stay constant from 8–10.
Increase linearly from 0–5, stay constant from 5–8, then decrease linearly from 8–10.
Decrease linearly from 0–5, stay constant from 5–8, then increase linearly from 8–10.
Increase linearly from 0–10 with no flat part.
Explanation
This question tests sketching graphs from qualitative descriptions (increasing/decreasing, linear/nonlinear, intervals) and analyzing graphs qualitatively describing behavior. Sketching: read description for behavior (increasing 0-5: up linear, constant 5-8: flat, decreasing 8-10: down linear), shape (linear pieces), no specific values but trends. For example, water amount increases to 5 min, constant to 8, decreases to 10 sketches as up-flat-down with straight lines. The correct sketch is increase linearly from 0–5, stay constant from 5–8, then decrease linearly from 8–10. Common errors include sketching decreasing when constant is described, or missing the flat part, or making parts curved. Sketching steps: (1) identify intervals and behavior (0-5: increasing linear, 5-8: constant, 8-10: decreasing linear), (2) determine shape (straight or flat lines), (3) plot turning points, (4) connect appropriately, (5) label axes (time, water amount). Mistakes: swapping increasing/decreasing intervals, drawing curves when linear, forgetting constant segment, wrong interval endpoints.
A student's distance from home (in miles) changes during a bike ride. The distance increases at a constant rate from 0 miles at 1:00 PM to 6 miles at 2:00 PM. Then the student rides back toward home at a constant rate and is 2 miles from home at 3:00 PM. Which sketch best matches this description (time on the x-axis, distance on the y-axis)?
A straight line increasing from (1 PM, 0) to (2 PM, 2), then increasing to (3 PM, 6).
A straight line increasing from (1 PM, 0) to (2 PM, 6), then a straight line decreasing to (3 PM, 2).
A curve that increases quickly at first and then levels off from 1 PM to 3 PM.
A straight line decreasing from (1 PM, 0) to (2 PM, 6), then increasing to (3 PM, 2).
Explanation
This question tests sketching graphs from qualitative descriptions (increasing/decreasing, linear/nonlinear, intervals) and analyzing graphs qualitatively describing behavior. Sketching: read description for behavior (increasing from 1-2 PM: graph goes up linearly, decreasing 2-3 PM: goes down linearly), shape (linear: straight lines), key points (starts at 0, to 6 at 2 PM, ends at 2 at 3 PM), draw accordingly (straight from (1 PM,0) up to (2 PM,6), straight down to (3 PM,2)). For example, distance starts at 0 miles at 1 PM, increases to 6 miles by 2 PM, then decreases to 2 miles by 3 PM sketches as up-then-down with linear pieces. The correct sketch is a straight line increasing from (1 PM, 0) to (2 PM, 6), then a straight line decreasing to (3 PM, 2). Common errors include sketching decreasing when description says increases, or making it curved when linear is specified, or wrong key points like ending at 6 instead of 2. Sketching steps: (1) identify intervals and behavior (1-2: increasing linear, 2-3: decreasing linear), (2) determine shape per interval (straight lines), (3) plot key points (start, peak, end), (4) connect with straight lines, (5) label axes (time, distance). Mistakes: confusing increasing with decreasing, drawing curves instead of straight, misreading rates or endpoints, wrong intervals.
A student sketches a function with these features:
- It starts at $(0,2)$.
- It increases at a constant rate until $x=4$.
- Then it stays constant (flat) from $x=4$ to $x=7$.
Which description matches the correct sketch?
A curve rising from (0,2) to (4,6), then continuing to rise but more slowly until (7,7).
A line segment falling from (0,2) to (4,-2), then a horizontal segment from (4,-2) to (7,-2).
A horizontal segment from (0,2) to (4,2), then a line segment rising from (4,2) to (7,6).
A line segment rising from (0,2) to (4,6), then a horizontal segment from (4,6) to (7,6).
Explanation
Tests sketching graphs from qualitative descriptions (increasing/decreasing, linear/nonlinear, intervals) and analyzing graphs qualitatively describing behavior. Sketching: read description for behavior (increasing constant rate to x=4, then constant), shape (linear then horizontal), key points ( (0,2), at x=4 some height, flat to x=7), draw accordingly (straight up to (4,height), then horizontal). For example, starts at (0,2), increases linearly to x=4, constant to x=7, like rising segment then flat. The correct sketch is a line segment rising from (0,2) to (4,6), then a horizontal segment from (4,6) to (7,6). A common error is making the increasing part horizontal or curving it, or falling instead of rising. Sketching tips: (1) identify intervals (0-4: increasing linear, 4-7: constant), (2) determine shape (linear, horizontal), (3) plot points, (4) connect appropriately, (5) label. Mistakes: wrong behavior per interval, adding curves.
A function starts at $(0,0)$. For $0\le x\le 4$, it increases linearly to $(4,8)$. For $4\le x\le 8$, it continues increasing but more and more slowly, leveling off near $y=10$.
Which sketch description matches this best?
A line from $(0,0)$ to $(4,8)$, then a curve that still rises but flattens toward $y=10$.
A line from $(0,0)$ to $(4,8)$, then a line decreasing to $(8,10)$.
A straight line from $(0,0)$ to $(8,10)$ with constant slope.
A curve that rises faster and faster after $x=4$ (gets steeper).
Explanation
This question tests sketching graphs from descriptions mixing linear and nonlinear behaviors, like switching from straight increase to slowing curve. Sketching involves plotting the linear part from (0,0) to (4,8) as a straight line, then a curve from there that rises but flattens toward y=10, indicating decreasing rate. For example, a function linear to x=4 then concave down approaching a horizontal level sketches as line then flattening curve. The correct description is a line from (0,0) to (4,8), then a curve that rises but flattens toward y=10, matching choice B. A common error is keeping it fully linear or making it steepen instead of flatten. To sketch: (1) identify intervals and behaviors (0-4: increasing linear, 4-8: increasing nonlinear slowing), (2) determine shapes (straight then curve), (3) plot key points like (0,0), (4,8), near (8,10), (4) connect with line then curve, (5) label axes. For analysis, match to options avoiding mistakes like adding decreases or wrong curving direction.
A student walks away from home, stops for a snack, then walks back home. Distance from home (y) versus time (x) should:
- increase at first,
- then stay constant for a while,
- then decrease back toward 0.
Which choice matches that situation?
Decrease, then stay constant, then increase
Increase, then decrease, then increase again
Stay constant the whole time
Increase, then stay constant, then decrease
Explanation
This question tests sketching graphs from qualitative descriptions of real-world scenarios, like distance changing with behaviors such as increasing, constant, then decreasing. Sketching involves mapping the description: distance increases (walking away), stays constant (stopping), then decreases (walking back), resulting in a rise, flat, then drop. For example, distance starting at 0, rising to some point, flat during snack, then falling back to 0 sketches as up-flat-down. The correct choice is increase, then stay constant, then decrease, matching choice C. A common error is reversing to decrease first or adding extra changes like increase-decrease-increase. To sketch: (1) identify intervals and behaviors (initial: increasing, middle: constant, end: decreasing), (2) determine likely linear shapes unless specified, (3) plot key points like start at 0, peak before flat, end at 0, (4) connect appropriately, (5) label with time (x) and distance (y). For analysis, match the sequence to options, avoiding mistakes like calling it constant overall.
A student sketches a function with these features:
- It starts at $(0,2)$.
- It increases at a constant rate until $x=4$.
- Then it stays constant (flat) from $x=4$ to $x=7$.
Which description matches the correct sketch?
A line segment rising from (0,2) to (4,6), then a horizontal segment from (4,6) to (7,6).
A line segment falling from (0,2) to (4,-2), then a horizontal segment from (4,-2) to (7,-2).
A horizontal segment from (0,2) to (4,2), then a line segment rising from (4,2) to (7,6).
A curve rising from (0,2) to (4,6), then continuing to rise but more slowly until (7,7).
Explanation
Tests sketching graphs from qualitative descriptions (increasing/decreasing, linear/nonlinear, intervals) and analyzing graphs qualitatively describing behavior. Sketching: read description for behavior (increasing constant rate to x=4, then constant), shape (linear then horizontal), key points ( (0,2), at x=4 some height, flat to x=7), draw accordingly (straight up to (4,height), then horizontal). For example, starts at (0,2), increases linearly to x=4, constant to x=7, like rising segment then flat. The correct sketch is a line segment rising from (0,2) to (4,6), then a horizontal segment from (4,6) to (7,6). A common error is making the increasing part horizontal or curving it, or falling instead of rising. Sketching tips: (1) identify intervals (0-4: increasing linear, 4-7: constant), (2) determine shape (linear, horizontal), (3) plot points, (4) connect appropriately, (5) label. Mistakes: wrong behavior per interval, adding curves.