16.1 Transformations of functions: Horizontal translations

Now that we have already learned about functions, we also need to learn about transformation.Graphs of different functions can be transformed in four ways, translation, reflection, stretching and compression. In this chapter we will be mostly talking about the first three.

In the first and second part of this chapter, we will learn about horizontal and vertical translations. In horizontal translations, the graph of the function is moved up or down, for every equation f(x) + b depending on the value of b. A positive value would indicate transforming the graph upwards, and a negative value would mean the transforming the graph downwards. For example, if we’re asked to graph x2+2x^2 + 2, we move the vertex of the graph of x2x^2, (0,0) two points upward, so now the vertex would be at (0,2).

For the vertical translations, we move the vertex of the graph of x2x^2 either to the left or to the right for ever equation f(x + b), depending again on the value of b. A positive value would mean to move the graph to the left, while a negative value would suggest moving the graph to the right. So in the case of f(x)=(x5)2f(x) = (x-5)^2, we move the vertex of x2x^2 5 units to the right at (5,0).

Apart from the vertical and horizontal translations, a graph could also undergo is also reflection.Reflection acts like mirrors. There are two kinds that you would find in graphing quadratics, Reflection across the y axis where y = f (-x) and Reflection across the x axis where y = -f(x).

Quadratic functions could also undergo stretching. Horizontal stretches are graphs of functions that appear to stretch away from the y axis, while the vertical stretches are graphs that appear to stretch away from the x axis.

In the last part of the chapter, we will look at the combined transformations. We will also look at coordinate mapping formula. After this chapter you will be able to understand more about transformation of graphs.

Transformations of functions: Horizontal translations

Horizontal translations refer to movements of a graph of a function horizontally along the x-axis by changing the x values. So, if y = f(x), then y = (x –h) results in a horizontal shift. If h > 0, then the graph shifts h units to the right; while If h < 0, then the graph shifts h units to the right.

Lessons

    • a)
      Sketch the following functions on the same set of coordinate axes:
      y=(x)2y = {\left( x \right)^2}      VS.      y=(x6)2y = {\left( {x - 6} \right)^2}      VS.      y=(x+5)2y = {\left( {x + 5} \right)^2}
    • b)
      Compared to the graph of y=x2y = {x^2}:
      • the graph of y=(x6)2y = {\left( {x - 6} \right)^2} is translated "horizontally" ________ units to the ______________.
      • the graph of y=(x+5)2y = {\left( {x + 5} \right)^2} is translated "horizontally" ________ units to the ______________.
    • a)
      y=f(x8)y = f\left( {x-8} \right)
    • b)
      y=f(x+3)y = f\left( {x+3} \right)
    • c)
      In conclusion:
      (x)(x8)\left( x \right) \to \left( {x-8} \right): shift __________ to the __________. All x coordinates ? ____________________
      (x)(x+3)\left( x \right) \to \left( {x+3} \right): shift __________ to the __________. All x coordinates ? ____________________
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Transformations of functions: Horizontal translations

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