Investigating transformations of quadratic graphs

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Trial Portfolio 1

XYZ

21st October 2012

Portfolio task – trail 1

1. Sketch the graphs of:

a. y = x2

b. y = x2 + 3

c. y = x2 – 2

What do you notice? Can you generalize?

The sketches above are graphs of quadratic functions and therefore parabolas. Although the shapes of all three graphs are the same, the graphs’ positions differ from one another. Taking the other two graphs relative to y = x2, we make observations. The graph of y = x2 + 3 shifts three units upwards and the graph of y = x2 – 2 shifts two units downwards relative to y = x2. The following table explains why this happens:

The table shows that for y = x2 + 3, three is added to the value of y (relative to y = x2). That means when, for example, x = 0, y = 3 ( x = 0, y = 0 for     y = x2). This explains the vertex of y = x2 + 3 moving three units upwards (the y value changes). The same principle applies to y = x2 – 2. In this case, two is subtracted from the value of y (relative to y = x2) and therefore the vertex (and the graph) shifts two units downwards (when x = 0, y = -2). Following are all three curves combined into one sketch:

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From the above observations, we can say that:

When |k| units are added to y, the graph shifts upward by k units.

When |k| units are subtracted from y, the graph shifts downwards by k units.

The phenomenon of the graph shifting upwards or downwards is called vertical translation.

        

2. Consider the graphs of:

a. y = x2

b. y = (x – 2)2

c. y = (x + 3)2

What do you notice? Can you generalize?

The sketches above are graphs of quadratic functions and therefore parabolas. Although the shapes of all three ...

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