\( \DeclareMathOperator{cosec}{cosec} \)

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

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

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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