SME
Grade 9 Math · Problem type
Average rate of change
Use this for “Find the average rate of change of f(x) = x² − 4x + 1 from x = 1 to x = 4.”
Chart
Secant
slope
f
(4) −
f
(1)
3
=
1
As h → 0 the secant turns into the tangent at P, slope −2
This is an example. Type your own values to start a new problem.
f(x)
=
a
x²
+
(
b
)
x
+
c
−
(
)
−
=
r
Δx
Change in input
worked out (not typed)
Formulas
y₁
=
a
×
x₁
2
+
b
×
x₁
+
c
−2
=
1
×
1
2
+
(−4)
×
1
+
1
y₂
=
a
×
x₂
2
+
b
×
x₂
+
c
1
=
1
×
4
2
+
(−4)
×
4
+
1
Δx
=
x₂
−
x₁
3 = 4 − 1
r
=
y₂
−
y₁
Δx
1
=
1 − (−2)
3
Assumptions
•
The average rate of change is the change in output over the change in input.
•
It is the slope of the line through the two points on the graph.
•
For a curve it changes with the interval; for a line it is always the slope.
Step-by-step
WE KNOW
a = 1, b = −4, c = 1, x₁ = 1, x₂ = 4
FIND
output at x₁, f(x₁) (y₁), output at x₂, f(x₂) (y₂), change in input (Δx), average rate of change (r)
Step 1 · Find output at x₁, f(x₁): y₁
Use y₁ = a × x₁² + b × x₁ + c
The output at the start of the interval.
y₁
=
1
×
1
2
−
4
×
1
+
1
y₁
=
1
−
4
+
1
y₁
=
−2
Step 2 · Find output at x₂, f(x₂): y₂
Use y₂ = a × x₂² + b × x₂ + c
The output at the end.
y₂
=
1
×
4
2
−
4
×
4
+
1
y₂
=
16
−
4
×
4
+
1
y₂
=
16
−
16
+
1
y₂
=
1
Step 3 · Find change in input (Δx)
Use Δx = x₂ − x₁
How far the input moves.
Δx = 4 − 1
Δx = 3
Step 4 · Find average rate of change (r)
Use r = (y₂ − y₁) ÷ Δx
The change in output over the change in input: the slope of the secant line.
r
=
1 − (−2)
3
r
=
3
3
r
=
1
Check
Put the numbers back in. Both sides should match.
−2
=
1
×
1
2
−
4
×
1
+
1
✓
1
=
1
×
4
2
−
4
×
4
+
1
✓
3 = 4 − 1 ✓
1
=
1 − (−2)
3
✓
Topic check
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