Translations and reflections of function graphs · Higher
| English | 中文 | Pinyin |
|---|---|---|
| graph translation/ɡræf trænˈsleɪʃn/ | 图像平移 | tú xiàng píng yí |
Does changing the input move the graph left or right?
- A graph starts with its minimum at (0,0). How does changing the input to x-3 move the minimum?
- This lesson studies graph translation 图像平移: Move every point of a graph by the same horizontal and vertical displacement.
Choose the mathematical structure
- y=f(x)+a shifts the output up by a. y=f(x-a) shifts the graph right by a. y=-f(x) reflects in the x-axis; y=f(-x) reflects in the y-axis. Transform the points as well as the formula, checking a known feature.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines graph translation?
Move every point of a graph by the same horizontal and vertical displacement.
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, y=f(x-3)+2=(x-3)²+2 has vertex (3,2). The point (1,1) on f moves to (4,3). At x=4 the new output is 3. The reflection y=-x² turns the minimum at the origin into a maximum. For a nonsymmetric graph, f(-x) mirrors each x-coordinate, not each y-coordinate.
Translations and reflections of function graphs
y=f(x)+a shifts the output up by a
Compare the model with the worked case and explain one change.
Find the vertex x-coordinate of (x-3)²+2.
The squared term is zero at x=3.
Test a tempting shortcut
- A positive number added inside the input, f(x+3), moves the graph left, not right. A reflection in the y-axis changes x; a reflection in the x-axis changes y.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
The graph y=f(x+3) always moves three units to the right. This claim is false. Explain which definition or assumption it violates.
Find its vertex y-coordinate.
At x=3, the output is 2.
The graph y=f(x+3) always moves three units to the right.
A positive number added inside the input, f(x+3), moves the graph left, not right. A reflection in the y-axis changes x; a reflection in the x-axis changes y.
Interpret a new situation
- AQA A13 Higher covers translations and reflections of a given function. Use features and transformed points to justify the sketch; do not substitute a geometric enlargement for this graph transformation.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find its output at x=4.
(4-3)²+2=3.
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
- 8300 · Higher · 3.2. 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.
Move every point of a graph by the same horizontal and vertical displacement. Choose the relationship, show the method, check its assumptions and interpret the result.