Algebra 1 · Unit 10
Describe scatterplots that need a piecewise rule or another curve, not a single line or parabola.
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Not a line, not a parabola
Some clouds rise then flatten, drop then rise, or follow two different trends on two sides of a cutoff.
Piecewise
One rule for x less than a number, another rule after. Example: a fee that is flat, then a per-mile rate.
Describe in pieces
Say what happens on the left, what happens on the right, and where it changes. Do not force one quadratic through a corner.
Other curves
Exponential growth, leveling off, or a U that is not symmetric can all appear. Name the shape you see.
Name the shape: two trends, or flattening.
For small x, y stays near 2 (flat).
After about x = 3, y rises linearly.
A piecewise description fits better than one line or one parabola.
Answer: flat, then rising
A line would keep rising. The data flatten. Describe two phases, or a leveling curve.
Answer: it flattens
Horizontal from 0 to 1 mile at y = 4, then a ray with slope 2.
Piecewise linear.
Answer: flat then slope 2
A parabola is smooth and U-shaped. A sharp V is two lines, piecewise.
Answer: piecewise V, not a parabola
Exponential.
Answer: exponential
Two groups. A piecewise or two-plot description, not one cloud.
Answer: two groups
Match a story to a piecewise or leveling shape.
Nearly horizontal, then a drop or a shallower slope after the cap.
Piecewise: one rate before the cap, another after.
Answer: piecewise at the cap
Rises quickly, then levels off near 8.
Not linear; not a symmetric parabola.
Answer: rises then levels
After 2 hours the rate changes. Piecewise: 3x for 0 ≤ x ≤ 2, then 6 + 1(x − 2) after.
Answer: rate changes at 2 hours
An S-shape (slow–fast–slow). A parabola would come back down; this may not.
Answer: S-shape / leveling
The peak is a change of story (enough sleep vs too much), not a symmetric U. Describe increase then decrease separately.
Answer: two pieces around 8 hours
Piecewise: linear before week 10, a different growth after.
One parabola through both parts would miss the corner.
Answer: piecewise at week 10
Choose among linear, quadratic, exponential, and piecewise.
A quadratic can look similar on a short window but will eventually bend the wrong way.
Piecewise linear names the corner (a launch week) and keeps a constant rate after.
If the story is “after launch we added a constant number each week,” piecewise linear matches the mechanism.
Answer: piecewise if there is a corner
Exponential doubling, then a cap (piecewise: exponential then constant).
Answer: exponential then flat
A quadratic is symmetric about the vertex. Unequal sides suggest two different pieces or a different curve.
Answer: not symmetric
n = 0: 20.
n = 1, 2, 3: still 20 (free visits).
n ≥ 4: 20 + 5(n − 3).
Answer: 20, then 20 + 5(n − 3) after 3 visits
Do not stop at the line. Try a quadratic, or split the domain if the U is a corner plus two trends.
Answer: try a curve or pieces
The real rule is defined in pieces at a cutoff. A parabola is smooth and would invent a curve the company does not use.
Answer: the rule is already piecewise
Example: download speed vs time of day — high at night, low at 8 p.m., not a parabola.
A periodic or piecewise daily pattern fits better than one U.
Answer: e.g. time-of-day pattern
A parabola can pass near a corner on a short window.
The fee’s mechanism is two lines. Residuals may still show a corner, and predictions past the window will curve.
Answer: wrong mechanism
Linear: straight cloud, constant first differences if binned.
Quadratic: U or hill, constant second differences.
Exponential: J-shape, roughly constant ratios.
Piecewise: a corner, a cap, or two groups with a cutoff in x.
Answer: see checklist
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