Lesson 4

In Try This 2 you obtained information about the graph of a quadratic function by looking at the parameters in the standard form and vertex form. You then verified the information gathered algebraically by graphing both forms of the function. If both the standard form and vertex form represent the same quadratic function, their graphs will be identical.

 

Self-Check 1

1. Consider the following pairs of equations. Does each pair represent the same function? Without graphing these functions, what information can you obtain from the parameters found in the standard and vertex forms? Graph your answer to confirm.

a.	(i) y = x2 − 6x − 3	(ii) y = (x − 3)2 − 12 Answer
b.	(i) y = x2 − 2x − 8	(ii) y = (x − 2)2 − 8 Answer
c.	(i) y = −2x2 − 2x + 2.5	(ii) y = −2(x + 2)2 + 3 Answer

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You may find Quadratics in Vertex Form - Activity A useful. This applet lets you graph a function in vertex form and see the standard form (labelled polynomial). (You may need to log in to LearnAlberta.ca first in order to run Quadratics in Vertex Form - Activity A.)

 

How do you see this applet being useful in Module 3? How could you have used the applet in Self-Check 1?

 

Example: Sketching the Graph of a Function in Standard Form

 

This example shows steps that could be used to sketch a graph of a quadratic function given in standard form.

 

Sketch the graph of y = 2x2 + 2x − 1.

 

Step 1: Look at parameter a in y = 2x2 + 2x − 1. You know that a is positive, so the parabola opens upwards. The value of a is greater than 1, so the parabola will be narrower than the graph of y = x2.

 

 

Step 2: Look at parameter c in y = 2x2 + 2x − 1. c is −1. Therefore, the y-intercept will be −1.

 

Step 3: Rearrange to vertex form, y = a(x − p)2 + q, by completing the square.

 

 

Step 4: Find the vertex from the parameters p and q. The vertex is found from (−p, q) and is .

 

Step 5: Look at parameter q.

 

 

From the value of q you know the minimum value of the function will be .

 

Step 6: Look at parameter p again.

 

 

This tells you that the axis of symmetry is at .

 

Step 7: Look at the axis of symmetry and the y-intercept. The axis of symmetry is at , so you know a second point will be the mirror image of the y-intercept (0, −1), at (−1, −1).

 

The y-intercept is very close to the vertex, so estimating the width of the graph could be tricky. Get a point further up the arm of the parabola. Try substituting 1 for x and find the point (1, 3), which is also on the graph.

 

Step 8: Sketch the graph.

 

 

Self-Check 2

1. Practise what you have learned about the properties of quadratic functions. Describe the characteristics of each function and, from those characteristics, sketch the graphs of the functions.
   
   
   1. y = 2x2 − 8x + 10 Answer
      
      
   2. Answer