9.3A Graphing quadratic functions (14 hours)

  •  characteristics and properties of quadratic graphs  
  • method of drawing quadratic graphs, relationship between coefficients of equation and graph on coordinate plane  
  • transforming quadratic graphs  
  • solve problems using quadratic graphs


 Subject learning objective

Learners can: 

formulate the rules of construction and construct graphics of functions 




 Subject Programme

Student will:

        

        influence the shape and position         of the curve;




         or 

        where a is not zero.


             




 Language learning objective

Learners can:

 understand and list orally the main steps and sub-steps involved in constructing quadratic graphs 





 Suggested teaching activities


  • Introduce unit by showing an equation to discuss for example
        

          and ask what learners notice,             then ask what type of equation           is it? 

  • Show by expansion that quadratic 
  • Recall work done in unit quadratic equations:
    • finding factors of quadratics solving by completing the square solving by formula




  • Challenge learners to solve a quadratic, choosing an appropriate method from those already encountered. 
  • Can anyone remember the type of graph that a quadratic produces? 
  • Ask learners to make a quick rough sketch and show each other, and discuss. 
  • Remind learners that this type of curve is a parabola, and it is one of the conic sections.


 
  • This unit will allow a detailed examination of this curve. 
  • Start by asking learners to sketch the graph of 
                       
 
         Is this quadratic? What are the            values of a, b, c?

  • Model the creation of a table of values, selection of scale on axes, plotting points, sketching the curve

  • Working with a partner, discuss the observable features and relate these to the equation: symmetry relation to 
    • y – axis 
    • relation to x – axis 
    • Where the curve changes direction

  • We understand the general form of the quadratic: 

  • Ask learners to predict what will happen if we change a, b or c How would we check the effects of changes? How many graphs would we need to confirm our thinking? Learners work as a group, deciding what to investigate, dividing the tasks between them, discussing findings, and preparing a presentation for other groups.

  • Hopefully, the following discoveries will be made: 
      1. The coefficient of x squared (a)
    • affects the shape of the curve (gradient) with higher coefficient producing a ‘tighter’ curve closer to the y-axis 
    • where a=0, the curve is a straight line (if b=0 as well, it is horizontal) 
    • negative coefficients invert the curve
                 2.  The coefficient of x      ,                         (b) alters the gradient of                        the straight line where                          x squared=0 (in effect,                          a linear equation)                                 Where a < > 0, it                                     ‘displaces’ the axis of                         symmetry left or right                           (in effect, the x -                                   squared   element                                 curves the line )

                   3.  The constant moves                             the curve up or down,                           and sets the point                                 where the curve                                     intercepts the y – axis.

  • Note the combined effect of all three: does this mean we can predict the curve of
                      
        Use the interactive graphing               software to examine the roots of          quadratics – where the curve              intercepts the x-axis produces            two roots, where the curve                  touches the x axis at one point            there is one root, curves which            do not meet the x axis have no            real roots

  • Learners to discuss in pairs what that means about the discriminant for these curves


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