A reliable method
Build a small table before using a graph. Compute output difference divided by input difference, with the same order in both. For g(x)=f(x-h)+k, track what happens to an existing point instead of guessing from the signs.
The common trap
The horizontal shift in f(x-h) is h units right, even though the expression subtracts h. An average rate is not the output itself.
Make the connection
A straight line has the same average rate on every interval. A nonlinear function usually does not, so always report the interval.
Compare rates and output differences
For f(x)=x², the interval [1,3] has output change 8 and input change 2, so average rate 4. On [3,5], the equal-length input interval has output change 16, so its average rate is 8. By contrast, L(x)=2x+1 has output change 4 on each interval and rate 2. The word average describes the two endpoints, not an average of the outputs. A student who computes (1+9)/2=5 has found a different quantity. Write the requested units beside the quotient: output units per input unit. A table can make the interval comparison easier to inspect than a memorised formula.
Transfer, explain and check
Use h(x)=3x²−2. Find its average rate on [1,3] and explain the effect of the vertical shift. Check: 12, because (25−1)/2=12; the −2 cancels from the output difference. The factor 3 triples the corresponding rate of x².
For f(x)=x², define g(x)=f(x − 3) + 4. What is g(6)?
- The inner input is 6 − 3 = 3.
- The original function squares that input: f(3)=9.
- Add the vertical shift 4: g(6)=13.
Compare the other answer choices
Skip the squaring operation.
Omit the vertical shift after squaring.
Add the vertical shift inside the square.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
AP Precalculus course and syllabus