Combined graph transformations and order
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| horizontal scale factor/ˌhɒrɪˈzɒntl skeɪl ˈfæktə/ | hệ số co dãn ngang |
Does moving it first give the same final graph?
- A compressed curve is moved to the right.
- This lesson studies horizontal scale factor 水平伸缩因子: The factor multiplying the original horizontal coordinates when a graph is stretched or compressed.
The method
- For g(x)=a f(b(x−h))+k with b≠0, an original point (u,v) maps to (u/b+h,av+k).
- Horizontal scale is 1/|b|, with reflection in the y-axis if b<0.
- Vertical scale is |a|, with reflection in the x-axis if a<0.
- Read the factored inner expression before naming a horizontal translation.
- The transformed domain follows u=b(x−h).
In g(x)=-2f(3x-6)+5, why is the horizontal translation 2 and not 6?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Transformations read from the FACTORED form a·f(b(x-h))+k: compression by 1/3 then shift right 2. Unfactored, the 6 hides the true translation - the orders do not commute.
A worked case
With f(x)=x², g(x)=−2f(3x−6)+5=−2f(3(x−2))+5=−18(x−2)²+5. Compress horizontally by factor 1/3, then translate right 2; reflect vertically and stretch by factor 2, then translate up 5. The points (−1,1),(0,0),(1,1) become (5/3,3),(2,5),(7/3,3). Thus the vertex is (2,5) and the range is y≤5. Compressing first and then shifting right 2 gives f(3(x−2)); shifting first and then compressing gives f(3x−2), whose vertex is at x=2/3 instead.
Combined graph transformations and order
For g(x)=a f(b(x−h))+k with b≠0, an original point (u,v) maps to (u/b+h,av+k)
Check the evidence and conditions behind a transformed or fitted function.
Find the vertex x-coordinate of g(x)=−2f(3x−6)+5 for f(x)=x².
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Factor the inner argument as 3(x−2); the original vertex u=0 maps to x=2.
Where it goes wrong
- The coefficient 3 inside f gives horizontal factor 1/3, not 3.
- In f(3x−6), the translation is 2, not 6.
- Translation and scaling generally do not commute.
- A symmetric example can hide a reflection: f(−x)=f(x) for x², but not for every f.
Not true: A horizontal translation and horizontal scaling can be applied in either order. Order matters: y=f(2(x+1)) and y=f(2x+1) are different graphs.
Find the maximum value of this g over all real inputs.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
g=−18(x−2)²+5, so the maximum is 5.
A horizontal translation and horizontal scaling can be applied in either order.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Order matters: y=f(2(x+1)) and y=f(2x+1) are different graphs.
Applying it
- Use point mapping to check every combined transformation, and transfer any restricted domain.
- If the original f is defined only for 0≤u≤3, the inner condition 0≤3(x−2)≤3 gives 2≤x≤3.
- Changing b to a negative value reverses endpoint order when solving the domain inequality.
- Plot the transformed reference points before joining the curve.
Evaluate g(1).
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
g(1)=−2(3−6)²+5=−18+5=−13.
Match each parameter of a·f(b(x-h))+k to its effect.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
A point (u,v) maps to (u/b+h, av+k); reading h before factoring b is the standard error.
In your course
- AQA A-level Mathematics (7357), specification section B.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Horizontal scale factor: The factor multiplying the original horizontal coordinates when a graph is stretched or compressed.
- For g(x)=a f(b(x−h))+k with b≠0, an original point (u,v) maps to (u/b+h,av+k).
- Watch for: the coefficient 3 inside f gives horizontal factor 1/3, not 3.