Quadratic functions II.
4th year of Secondary education. Option A.
 
Drawing the graph of the function y = x2 + bx + c.

We are going to look at the graph of the function y=(x+h)2+k and how any function of the equation y = x2 + bx + c can be transformed into another equation of the form y = (x + h)2 + k.

Give h and k different values and write the functions for the graphs into your notebook along with their corresponding vertices.

Change the initial values of h given in the window until h =0 and then do the same for k. Look carefully at how the graph in the window first moves along the x-axis and then along the y-axis. It does this until it coincides with the graph of y=x2. 

 

 

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

Expand the following equation, y=(x+h)2 + k; first applying Newton's binomial expansion and then the rest. Make this equation equal to y = x2 + bx + c and you should reach the following conclusion:

b = 2h, so h = b/2

c = h2 + k, so k = (4c - b2 )/4

Use this information to change the parabola y=x2+4x+5 to a different one with the form y=(x+h)2+k. This should make it easier to draw graphs by using translations along the x-axis and the y-axis of the graph y=x2.

Drawing graphs of the function y = ax2 + bx + c.
The graphs for y = 3x2 and y = -5x2  are illustrated in the following window. Compare them to the graph of y = x2. How do they compare to this graph. Is their shape more open or closed? Are they vertex down or vertex up?

Try to write a general conclusion (given that k and h are equal to 0 and a has different values) In the functions y=ax2 where a>1, the graphs are vertex .... and more closed than in the graph of y = x2.

Write a similar sentence for 0<a<1and a<0.

 

 

Draw the graphs of the following functions:

y=3x2+5 (you need to use the arrows next to each parameter to make a=3, h=0, k=5.) Write down the values of its vertex.

y=-5x2+2.Write down the values of its vertex.

Compare them to the graphs of the previous functions y=3x2 and y=-5x2 (changing h and k until they are equal to 0).

Draw graphs of the following functions in the same way (Write down the values for their vertices in your notebook):

y=3(x-1)2+5 (a=3, h=-1,k=5)

y=-5(x+2)2+2

 

 

Write a conclusion in your notebook (about the vertices of these parabolas, their shape etc.)

Now expand the following equation y=a(x+h)2+k. Make it equal to y=ax2+bx+c and you should reach the conclusion:

b= 2ah, so h=b/2a

c=ah2+k, so k=(4ac-b2)/4a

 

 

 

Use this information to transform the parabola y=3x2+12x+17 into an equation of the function y=a(x+h)2+k. Now it will be easier to draw the graph by translating the graph of y=3x2 along the x-axis and y-axis.

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