Translations and reflections of function graphs · Higher
도입됨| English | 의미 |
|---|---|
| graph translation/ɡræf trænˈsleɪʃn/ | 그래프 평행이동 |
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.
The method
- 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.
Why does y=f(x+3) move the graph LEFT by 3?
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
Track one point: the feature at input 2 on f appears when x+3=2, i.e. at x=-1 - three units left. Changes INSIDE the bracket act in the opposite direction to intuition.
A worked case
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.
난이도 2/5 · 140 코인 + 오늘 첫 완료 시 소액 보너스
The squared term is zero at x=3.
Where it goes wrong
- 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.
Not true: The graph y=f(x+3) moves three units to the right. f(x+3) moves the graph three units to the left.
Find its vertex y-coordinate.
난이도 2/5 · 140 코인 + 오늘 첫 완료 시 소액 보너스
At x=3, the output is 2.
The graph y=f(x+3) moves three units to the right.
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
f(x+3) moves the graph three units to the left.
Applying it
- 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.
Find its output at x=4.
난이도 2/5 · 140 코인 + 오늘 첫 완료 시 소액 보너스
(4-3)²+2=3.
Match each transformation of y=f(x) to its effect.
난이도 2/5 · 140 코인 + 오늘 첫 완료 시 소액 보너스
Outside changes move the graph vertically; inside changes act horizontally and in the opposite-signed direction.
In your course
- AQA GCSE Mathematics (8300), Higher tier, specification section 3.2.
- Paper 1 is non-calculator: keep fractions, surds and π exact and show each step. On calculator papers, still write the expression before its value.
- Graph translation: Move every point of a graph by the same horizontal and vertical displacement.
- 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.
- Watch for: A positive number added inside the input, f(x+3), moves the graph left, not right.