Lesson 2

Suppose you are given the vertex of a parabola (6, 8) and one of the x-intercepts (2, 0). From this information, you can determine the following:

• the coordinates of the other x-intercept
  
  
• whether the value of a in the quadratic function is negative or positive
  
  
• the maximum or minimum value of the function

This information can be determined without drawing the graph by using the axis of symmetry and some reasoning. Since the vertex is at (6, 8), the axis of symmetry is at x = 6. The given x-intercept (2, 0) is 4 units to the left of the axis of symmetry (6 − 2 = 4). Therefore, the other x-intercept will be 4 units to the right side of the axis of symmetry at x = (6 + 4), or x = 10.

 

Remember that the value of a in the quadratic function y = ax2 + bx + c indicates whether the parabolic graph of the quadratic function opens upwards or downwards. If a is positive, the graph opens upwards and the vertex is the minimum value of the function. If a is negative, the graph opens downwards and the vertex is the maximum value of the function. The given x-intercept (2, 0) is below the vertex (6, 8); so the graph must open downwards. This tells us the value of a in the quadratic function is negative and the vertex is a maximum.

 

Therefore, the maximum value of the function is 8, the value of y at the vertex.

 

Self-Check 3

1. Use the following graph to complete this question.
   
   
   

   For the graph, describe the
   
   

   1. vertex
      
      
   2. direction of opening
      
      
   3. axis of symmetry
      
      
   4. domain
      
      
   5. range
      
      
   6. x-intercepts
      
      
   7. y-intercept
      
      
   8. maximum or minimum value of the function
   
   
   Answer
   
   
2. Use the following graph to complete this question.
   
   
   
    
   For the graph, describe the
   
   

   1. vertex
      
      
   2. direction of opening
      
      
   3. axis of symmetry
      
      
   4. domain
      
      
   5. range
      
      
   6. x-intercepts
      
      
   7. y-intercept
      
      
   8. maximum or minimum value of the function
   
   
   Answer
   
   
3. Using the principles of symmetry, sketch the graph of y = −2x2 +16x − 16. Use the following information to sketch your graph:
   
   
   • The vertex is at (4, 16).
   • One point on the graph is (2, 8).
   • Another point on the graph is (0, −16).
     
     
   If you need some graph paper to complete this question, you can print out a grid at 1-cm Grid Paper. If you want to sketch the graph electronically and know how to use a drawing program, feel free to do it that way.
   
   Answer

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Did you notice that the y-intercept is the same as the value of c in the standard form of the quadratic function y = ax2 + bx + c? Look back at the graphs in Self-Check 3. Observe how the constant at the end of the function is always the y-intercept. That will always be true.

 

To see why, put the value of 0 in for x in the standard form of the quadratic function.

 

y = ax2 + bx + c

y = a(0)2 + b(0) + c

 

Can you see that for a quadratic function in standard form, the y-intercept will be y = c?

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