Quadratic functions I.
4th year of Secondary education. Option A.
 
Drawing the graph of the function y = x2.
In your exercise book take different values for x and work out their squares.

If we plot these values on a set of axes we get an approximate graph of their function. This is made easier by using the Descartes programme.

Compare the shape of the graph you've drawn in your notebook with the one you get in the window. Reduce the scale to 5 and change the values of x again. 

Click on the Start button and increase the scale to 150. Change the x-coordinate values again from negative values through to positive values.

Indicate the points between the ones already given, e.g. 0.05, 0.15, -0.05, -0.15 etc. 

Make the values of x greater than and less than 0 by using the arrows. (You can also write a particular value for x by deleting the present value).

Drawing the graph of the function y = x2 + k. 

You should be able to work out how to draw the graphs of functions such as y=x2+3, (where k = 3), y=x2+7, (where k = 7) and y=x2-4, (where k = -4) by using the function in the last activity as an example.

To draw the graphs for the functions above in the window, first put in the value for k (change the scale if necessary so that you can see the graph clearly).

Write down the vertices (minimum points) of the curves in your notebook.

Try and work out how these coordinates can be obtained just by looking at the equation. Complete the following sentence to help you:

 

The function y = x2 + k is a translation of the points for y = ....... along the ...... axis. The vertex of y = x2 + k is (...,...).

Draw the graphs in your notebook, without giving the x values and following the model for y = x2  for the functions y = x2 - 5 and y = x2 + 9.

Drawing the graph of the function y = (x + h)2.
We are going to draw graphs of the different functions of y = (x + h)2 for different values of h. This time, instead of being a series of points the graphs are continuous lines and we should note that the shape of the graph is the same as the graph for y=x2 . The only difference is a series of translations along one of the axes.

You can change the values for k and h by using the arrows or by writing in their values. Change the scale until you can see the whole graph clearly.

Give positive and negative values for h. Write the different functions for y=(x+h)2 in your notebook as you draw the graphs in the window using the programme and write down the coordinates of the vertices of each graph.

By now, you should know that the initial values for k and h are 3 and -5 respectively. Use the arrows to change the values so that they are both equal to zero. You should notice that by changing these initial values all you are doing is translating the graphs along the axes. The graph finally coincides with the graph for y=x2 as they are the same shape.

 

 

Complete the following sentence: The function y = (x + h)2 is a translation of the points for y =.... along the ...... axis. The vertex of y = (x + h)2 is (...,...).

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  Carlos-Vidal Díaz Vicente
 
Spanish Ministry of Education. Year 2001