Combined graph transformations and order
| English | Português |
|---|---|
| horizontal scale factor/ˌhɒrɪˈzɒntl skeɪl ˈfæktə/ | horizontal scale factor |
A compressed curve is moved to the right. Does moving it first give the same final graph?
- A compressed curve is moved to the right. Does moving it first give the same final graph?
- 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.
Choose the mathematical structure
- 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).
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines horizontal scale factor?
The factor multiplying the original horizontal coordinates when a graph is stretched or compressed.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
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.
Factor the inner argument as 3(x−2); the original vertex u=0 maps to x=2.
Test a tempting shortcut
- 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.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
A horizontal translation and horizontal scaling can always be applied in either order. This claim is false. Explain which definition or assumption it violates.
Find the maximum value of this g over all real inputs.
g=−18(x−2)²+5, so the maximum is 5.
A horizontal translation and horizontal scaling can always be applied in either order.
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.
Interpret a new situation
- 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.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Evaluate g(1).
g(1)=−2(3−6)²+5=−18+5=−13.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 7357 · A-level · B. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
The factor multiplying the original horizontal coordinates when a graph is stretched or compressed. Choose the relationship, show the method, check its assumptions and interpret the result.