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Analyze & Sketch
Function Graphs

Learn to read increasing, decreasing, linear, and nonlinear behavior from a graph, then sketch your own from a verbal description.

8th Grade Math

What Is a Function Graph?

A function graph is a picture that shows the relationship between two quantities: an input (the horizontal axis, \( x \)) and an output (the vertical axis, \( y \)). Every point on the curve tells you: "When the input is this, the output is that."

Input (\( x \)) Output (\( y \)) input output

Increasing vs Decreasing

Increasing
As you move right along the \( x \)-axis, the graph goes up. The output gets larger.
Decreasing
As you move right along the \( x \)-axis, the graph goes down. The output gets smaller.
Increasing Decreasing

Linear vs Nonlinear

Linear
The graph is a straight line. The rate of change is constant: the output changes by the same amount for each unit of input.
Nonlinear
The graph curves. The rate of change varies: sometimes the output changes quickly, sometimes slowly.
Linear Nonlinear

The Four Behaviors at a Glance

Linear Increasing Nonlinear Increasing Linear Decreasing Nonlinear Decreasing

Reading a Graph: the Core Rule

To describe a function qualitatively, scan from left to right and answer two questions for each section of the graph:

1
Direction: Is the graph going up (increasing) or down (decreasing)?
2
Shape: Is it a straight line (linear) or a curve (nonlinear)?

Constant vs Zero Output

Constant (Flat)
The graph is a horizontal segment. The output doesn't change. It could be at any height, not just \( y = 0 \).
Output = 0
The graph touches or crosses the \( x \)-axis. This means the output value is zero at that specific input.
Constant (flat at any height) output = 0 Crosses x-axis

How to Analyze a Graph

1
Identify the axes: What does \( x \) represent? What does \( y \) represent?
2
Scan left to right: Break the graph into sections wherever the behavior changes.
3
Label each section: increasing/decreasing/constant, and linear/nonlinear.
4
Note key points: Where does the graph start, stop, reach its highest/lowest value, or cross the \( x \)-axis?
5
Write it in words: Describe the full story of the function from beginning to end.

How to Sketch from a Description

1
Draw and label axes: Write the two quantities on the \( x \)- and \( y \)-axes.
2
Plot the starting point: Where does the function begin?
3
Sketch each section: Use a ruler for linear parts, a freehand curve for nonlinear parts. Go up for increasing, down for decreasing, flat for constant.
4
Mark key features: Label any peaks, valleys, or where the graph crosses an axis.
5
Check your sketch: Re-read the description and verify every detail matches.

Example 1: Reading a Graph

A ball is thrown into the air. The graph below shows its height (in feet) over time (in seconds).

Time (seconds) Height (feet) peak increasing decreasing 0 5

Example 2: Multiple Sections

The graph shows the speed of a car during a 10-minute trip. Describe the behavior in each section.

Time (minutes) Speed (mph) A B C 0 3 6 10
A
Linear increasing: the car speeds up at a steady rate from 0 to 3 min.
B
Constant: the car maintains the same speed from 3 to 6 min.
C
Nonlinear decreasing: the car slows down, gradually at first, then more steeply, reaching 0 at 10 min.

Example 3: Sketch from Words

Weeks Height (cm) slow start steady growth stops growing

Each keyword in the description maps to a specific graph feature: "slowly at first" means a gentle curve, "steady rate" means a straight line, and "stops" means flat.

Example 4: Temperature Over a Day

Describe the temperature graph below.

Hour of the Day Temp (°F) peak: ~3pm 6am noon 10pm

Common Mistakes to Avoid

✕ Confusing Slope with Height
"The graph is high, so it's increasing." No: increasing means the graph is going up as you move right, not just that the \( y \)-value is large.
✕ Ignoring Sections
Describing the entire graph as "increasing" when it only increases in part. Always break the graph into sections and describe each one.
✕ Calling Every Curve "Nonlinear Increasing"
A curve that goes down is nonlinear decreasing. "Nonlinear" describes the shape; "increasing/decreasing" describes the direction.

Summary Table

Feature What to Look For Key Words
Increasing Graph rises left to right goes up, grows, rises
Decreasing Graph falls left to right goes down, drops, falls
Constant Graph is flat (horizontal) stays the same, levels off
Linear Straight line steady rate, constant rate
Nonlinear Curved line speeds up, slows down, curves

Question 1 of 5

A graph of distance vs. time shows a straight line going upward from left to right. Which description best fits this section?

Question 2 of 5

A graph shows a curve that rises steeply at first, then levels off into a flat horizontal line. Which sequence of behaviors describes this graph?

Question 3 of 5

Time Value

Which description best matches the graph above?

Question 4 of 5

You need to sketch a graph for: "The water level rises quickly at first, then more and more slowly, and finally stops rising." Which graph shape should you draw?

Question 5 of 5

A student says: "The graph is high on the left side, so the function is increasing there." Is the student correct?