Lesson 4

Lesson 4 Summary

 

In this lesson you investigated the following questions:

• How do you determine the characteristics of a quadratic function in the form y = ax2 + bx + c?
  
  
• How can you determine whether two functions written in the forms y = ax2 + bx + c and y = a(x − p)2 + q represent the same function?

© sss78/31470934/Fotolia

 

You learned that some characteristics of the graph of a quadratic function can be determined directly from the standard form of the quadratic function.

• The y-intercept will always be the value of c in the function y = ax2 + bx + c.
  
  
• The value of a is the same in either form of the quadratic function. The value of a influences the width of the curve of the parabola and whether the curve will open downward or upward. If a is negative, the parabola opens downward; if a is positive, the parabola opens upward.

To determine the axis of symmetry and the coordinates of the vertex of the parabola, it is useful to convert the function to the vertex form by completing the square. In the vertex form, y = a(x − p)2 + q, the following are true:

• The vertex is at (p, q).
  
  
• The axis of symmetry is at x = p.
  
  
• The minimum value if a > 0 or the maximum value if a < 0 occurs at y = q.

You learned three ways to check whether a function given in the form y = ax2 + bx + c represents the same function as one given in the form y = a(x − p)2 + q:

• Method 1: Graph both forms of the quadratic function.
  
  
• Method 2: Square the binomial in the vertex form, y = a(x − p)2 + q, and simplify to see if the function matches the standard form.
  
  
• Method 3: Complete the square of standard form, y = ax2 + bx + c, to see if the function matches the vertex form of the function.

In the next lesson you will learn to model situations using quadratic functions. You will solve problems by analyzing the functions.