International Baccalaureate Mathematics

Syllabus Content

Transformations of graphs. Translations: y=f(x)+b;y=f(x-a). Reflections (in both axes): y=-f(x);y=f(-x). Vertical stretch with scale factor p: y=pf(x). Horizontal stretch with scale factor 1/q: y=f(qx). Composite transformations.

Here are some specific activities, investigations or visual aids we have picked out. Click anywhere in the grey area to access the resource.

Graph Patterns Find the equations which will produce the given patterns of graphs.

Transformations of Functions A visual aid showing how various transformations affect the graph of a function.

Here are some exam-style questions on this statement:

"The graph of the curve A with equation $y=f(x)$ is transformed to give the graph of the curve B with equation $y=5-f(x)$." ... more

"(a) By completing the square, solve $x^2+8x+13=0$ giving your answer to three significant figures." ... more

"The graph of the following equation is drawn and then reflected in the x-axis" ... more

"(a) Find the interval for which $x^2 - 9x + 18 \le 0$" ... more

"(a) Write $2x^2+8x+27$ in the form $a(x+b)^2+c$ where $a$, $b$, and $c$ are integers, by 'completing the square'" ... more

"Let $f (x)=a(x-b)^2+c$. The vertex of the graph of $f$ is at (4, -3) and the graph passes through (3, 2)." ... more

"A function is defined as $f(x) = 2{(x - 3)^2} - 5$ ." ... more

"$f$ and $g$ are two functions such that $g(x)=3f(x+2)+7$." ... more

"Let $f(x)=5x^2-20x+k$. The equation $f(x)=0$ has two equal roots." ... more

"Let $f(x) = {x^2}$ and $g(x) = 3{(x+2)^2}$ ." ... more

"Part of the graph of a function, $f$ , is shown in the following diagram. The graph of $y = f(x)$ has a y-intercept at $(0, 1.5)$ , an x-intercept at $(a , 0)$ and horizontal asymptotes $y = 5$ and $y = -2$." ... more

"Let $f$ and $g$ be functions such that $g(x) = 3f(x - 2) + 7$ ." ... more

"The graph of $ y = f(|x|) $ for $ -2 \leq x \leq 2 $ is shown in the following diagram." ... more

"Two functions are defined as follows: $f(x) = 2\ln x$ and $g(x) = \ln \frac{x^2}{3}$." ... more

See all these questions

Click on a topic below for suggested lesson Starters, resources and activities from Transum.

Graphs This topic includes algebraic and statistical graphs including bar charts, line graphs, scatter graphs and pie charts. A graph is a diagram which represents a relationship between two or more sets of numbers or categories. The data items are shown as points positioned relative to axes indicating their values. Pupils are typically first introduced to simple bar charts and learn to interpret their meaning and to draw their own. More sophisticated statistical graphs are introduced as the pupil's mathematical understanding develops. Pupils also learn about coordinates as a pre-requisite for understanding algebraic graphs. They then progress to straight line graphs before learning to work with curves, gradients, intercepts, regions and, for older pupils, calculus.

Furthermore

Official Guidance, clarification and syllabus links:

Students should be aware of the relevance of the order in which transformations are performed.

Dynamic graphing packages could be used to investigate these transformations.

Example: Using $y=x^2$ to sketch $y=3x^2+2$

Link to: composite functions (SL2.5).

Not required at SL: transformations of the form $f(ax+b)$.

In the study of functions, understanding graph transformations is crucial for interpreting and predicting the behaviour of different equations. A transformation modifies the appearance of the original graph, which can be translated, reflected, stretched, or compressed.

Translations involve shifting the graph without changing its shape. The function $ y = f(x) + b $ results in a vertical translation by $ b $ units. If $ b $ is positive, the shift is upwards; if negative, the shift is downwards. Conversely, $ y = f(x + a) $ translates the graph horizontally by $ a $ units to the left for positive $ a $, and to the right for negative $ a $.

Reflections flip the graph over a specific axis. Reflecting over the x-axis, $ y = -f(x) $, inverts the graph upside-down. Reflecting over the y-axis, $ y = f(-x) $, produces a mirror image of the graph.

Vertical stretches are achieved by multiplying the function by a scale factor $ p $, denoted as $ y = pf(x) $. The graph stretches away from the x-axis if $ p > 1 $ or compresses towards it if $ 0 < p < 1 $.

Horizontal stretches are expressed by $ y = f(qx) $, where $ 1/q $ is the scale factor. The graph stretches horizontally away from the y-axis for $ q < 1 $ and compresses towards it for $ q > 1 $.

Composite transformations involve performing more than one transformation on a graph. It's essential to note the order in which transformations are applied, as it sometimes affects the outcome.

Let's consider an example. To apply a vertical stretch by a factor of 3 combined with a horizontal shift 2 units to the right, the transformation would be represented as:

$$ y = 3f(x - 2) $$

This indicates that every point on the graph of $ y = f(x) $ is first moved 2 units to the right, and then stretched vertically away from the x-axis by a factor of 3.

This video on Transformation of Functions is from Revision Village and is aimed at students taking the IB Maths AA Standard level course

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