Grade 8 Sketching and Describing Function Graphs

Visual Model 1

Question: A toy platform follows the height-time graph shown. Which description is accurate?

• A. The object goes up constantly, then down, then flat
• B. The object stays level, drops, stays level, rises, stays level
• C. The object increases, then decreases throughout
• D. The object decreases in height initially

Why it works: From 0 to 2s: flat (height constant at 4m). From 2 to 3s: the height drops to 1m. From 3 to 4s: flat at 1m. From 4 to 5s: the height rises to 3m. From 5 to 6s: flat at 3m.

Answer: The graph shows constant, decreasing, and increasing segments

Visual Model 2

Question: A student saves money over time. The graph shows: increases for 3 weeks, then constant for 3 weeks. The \(y\)-intercept is 1. Which is true?

• A. The student had $0 at the start
• B. The student had $1 initially and saves $1 per week, then stops
• C. The student spends money after week 3
• D. The function is linear over all 6 weeks

Why it works: The \(y\)-intercept (at \(x=0\)) is 1, so the student has $1 initially. From 0 to 3 weeks, the graph rises linearly from 1 to 4, a gain of $1 per week. From week 3 to 6, the graph is flat at 4, so no additional savings.

Answer: The student starts with $1 and saves at a constant rate initially

Example 1

Question: Identify the intervals where the function is increasing.

• A. From \(x=1\) to \(x=3\) only
• B. From \(x=-1\) to \(x=1\)
• C. From \(x=1\) to \(x=3\) and from \(x=3\) to \(x=5\)
• D. From \(x=-1\) to \(x=3\)

1. From \(x=-1\) to \(x=1\): the function is constant at \(f(x)=2\) (neither increasing nor decreasing).
2. From \(x=1\) to \(x=3\): the function increases from 2 to 4.
3. From \(x=3\) onward: the function decreases.

Answer: The function increases from \(x=1\) to \(x=3\)

Example 2

Question: A traveler's distance graph shows: 0 to 2 hours rising, then a jump, then flat. What does the jump represent?

• A. The traveler rested without moving
• B. The traveler waited at the same distance
• C. This graph cannot represent a real physical scenario (discontinuity)
• D. The traveler went backward in distance

1. A jump discontinuity in a distance-time graph would mean the distance changes suddenly without showing the travel in between, which is not physically realistic for normal motion.
2. A real distance-time graph should change continuously.

Answer: A discontinuous jump indicates an impossible real-world scenario

Example 3

Question: Which graph matches: "A ball is thrown upward and falls back down"?

• A. A line going up and to the right
• B. A curve (parabola) that rises then falls
• C. A flat horizontal line
• D. A line going down and to the right

1. The height increases initially (upward throw) then decreases (falling back).
2. A parabola captures this curved motion better than a straight line.
3. Option A is a line, C is constant, and D is always decreasing.

Answer: A parabola that opens downward represents motion under gravity

Problem 1

Question: An item's price over 8 weeks is shown. During which week does the price increase the fastest?

• A. Weeks 1-3
• B. Week 3
• C. Weeks 3-5
• D. Weeks 5-8

Why it works: From week 3 to 5, the price jumps from $1 to $4 in 2 weeks, a rate of $1.50/week. All other segments are flat.

Answer: The steepest slope occurs from weeks 3 to 5

Problem 2

Question: A runner's speed increases for the first 5 minutes, stays constant for 10 minutes, then decreases as they cool down. Which graph shape represents this? \StoryGraph{(0,1)--(2.5,5)--(6,5)--(8,2)}

• A. Three straight segments: rising, flat, falling
• B. A curve that rises steeply, then flat, then falls
• C. A U-shaped parabola
• D. A line with constant positive slope

Why it works: The scenario describes three distinct behaviors: rising speed (increasing), constant speed (flat), and decreasing speed (falling). This is a piecewise linear graph with three segments.

Answer: Three linear segments: increase, constant, decrease