Rates of Change in Linear and Quadratic Functions
Introduced| English |
|---|
| quadratic function/kwɒˈdrætɪk ˈfʌŋkʃn/ |
| linear function/ˈlɪnɪə ˈfʌŋkʃn/ |
| constant/ˈkɒnstənt/ |
| slope/sləʊp/ |
| average rate of change/ˈævrɪdʒ reɪt ɒv tʃeɪndʒ/ |
Two kinds of climb
- An escalator lifts you at a steady pace: every second gains the same height.
- A jet on the runway is different: each second it gains more speed than the last.
- Both are "increasing", but the way they change is completely different.
- Linear and quadratic functions are the first two members of this story.
A linear function: constant rate
- A linear function 线性函数 graphs as a straight line.
- Its slope 斜率 is the same everywhere, so its average rate of change 平均变化率 is the same over any interval.
- Over $[0, 2]$ or $[10, 12]$ or any other stretch — you always get the identical number.
- That single unchanging rate is what "linear" really means.
The average rate of change of a linear function over any interval is…
A straight line has one fixed slope, so the average rate of change is the same over every interval — that is what makes it linear.
A quadratic function: a changing rate
- A quadratic function 二次函数 graphs as a curved parabola.
- Because it bends, its slope is different at different places — the rate of change keeps changing.
- Near the vertex it is nearly flat; far from the vertex it is steep.
- So its average rate of change depends on which interval you pick.
One slider turns a parabola into a straight line
y = ax² + bx + c
Set a = 0 to get a straight line — its slope is constant. Make a ≠ 0 and the curve bends, so now the rate of change keeps changing.
For a quadratic function, the average rate of change is the same over every interval.
A parabola bends, so its slope keeps changing — different intervals give different average rates of change.
Equal steps reveal the pattern
- Take equal steps in the input and list the outputs of $y = x^2$: $0, 1, 4, 9, 16$.
- The jumps between them — the first differences — are $1, 3, 5, 7$.
- Those jumps are not constant, but they grow by the same amount ($2$) every time.
- So a quadratic's rate of change is not fixed, yet it changes in a perfectly steady, linear way.
For $y = x^2$, the first differences at $x = 1, 2, 3$ are $3, 5, 7$. By what constant amount do they grow each step?
$5 - 3 = 2$ and $7 - 5 = 2$. This constant second difference is the fingerprint of a quadratic.
For a quadratic, equal steps in $x$ give first differences that…
The first differences themselves form a linear pattern — they grow by the same amount each step. (Doubling each time would be exponential.)
Every average rate is a secant slope
- Whichever function you have, the average rate of change over $[a, b]$ is the slope of the secant line through its endpoints.
- For a line, every secant lies right on top of the graph — same slope every time.
- For a parabola, each secant tilts differently — that is the visible sign of a changing rate.
- Reading secant slopes is how you compare how two functions change.
Put the steps for finding an average rate of change over $[a, b]$ in order.
Outputs first, then their difference, then divide by the input difference — that is the secant slope.
Select all true statements.
A quadratic's rate is not constant — it changes steadily (its second difference is constant). The other three statements are true.
Do not call a quadratic's rate "constant" just because the pattern is predictable. The rate itself changes from interval to interval; it is the change in the rate that is constant. For a straight line, that change in the rate is zero.
A dropped ball falls a distance $d = 5t^2$ metres after $t$ seconds.
- Distances at $t = 0, 1, 2, 3$: $0, 5, 20, 45$ metres.
- First differences (distance each second): $5, 15, 25$ — growing by a constant $10$.
- The ball's speed rises linearly even though the distance is quadratic.
A linear function changes at one constant rate — every secant has the same slope. A quadratic function's rate of change is not constant: over equal steps its first differences grow by a fixed amount, so the rate changes steadily. The average rate of change over $[a, b]$ is always the secant slope.