Lesson 4

So, how does changing the values of a, h, and k in a quadratic function written in vertex form, y = a(x − h)2 + k, affect the shape and position of the parabola? The answer to this question can help determine the characteristics of a quadratic function written in vertex form.

 

Try This 1

 

Use the applet Quadratic Function: Vertex Form to investigate how changing the values of a, h, and k for a quadratic function written in vertex form, y = a(x − h)2 + k, affects the shape and the position of the parabola.

 

 

Complete the following questions.

1. Set the values of h and k to zero. Set the a slider to 1. You should have the graph of the function
   y = x2. Use the a slider to change the coefficient of x2. Try both positive and negative values
   for a.
   
   
   1. How do the parabolas change as you change the value of a?
      
      
   2. For each function you graphed in part a, determine the coordinates of the vertex and the equation of the axis of symmetry.
      
      
2. Set the slider for a to +1, and set the value of k to 0. Use the h slider to investigate how changing the value of h in the quadratic function y = a(x − h)2 + k affects the position of the graph. Try both positive and negative values for h.
   
   
   1. How do the parabolas change as you change the value of h?
      
      
   2. How do the coordinates of the vertex and the equation of the axis of symmetry change as you change the value of h?
      
      
3. Set the slider for a to +1, and set the value of h to 0. Use the k slider to see how changing the value of k in the quadratic function y = a(x − h)2 + k affects the position of the graph. Try both positive and negative values for k.
   
   
   1. How do the parabolas change as you change the value of k?
      
      
   2. How do the coordinates of the vertex and the equation of the axis of symmetry change as you change the value of k?
      
      
4. Make a conjecture about how the values of a, h, and k determine the characteristics of a parabola.
   
   
5. Test your conjecture. Drag the sliders for a, h, and k to create the quadratic functions shown in the Vertex Table. For each function, determine the coordinates of the vertex and the equation of the axis of symmetry.

Save your completed Vertex Table to your course folder.

 

Share 1

 

Based on your observations from Try This 1, discuss the following questions with another student or appropriate partner.

1. How does the position of the graph change as the value of h changes? Describe the pattern you see.
   
   
2. How does the position of the graph change as the value of k changes? Describe the pattern you see.
   
   
3. Why do you suppose this form of the quadratic function is called the vertex form?