Example 1: Quadratic Expression Representing a Function
a. A quadratic function is defined by g(x) = 2x2 + 12x + 1. Write this in the completed-square (vertex) form and show all the steps.
b. Where is the vertex of the graph of this function located?
c. Look at the completed-square form of the function. Can you name the parent function? How do you know?
d. What transformations have been applied to the parent function to arrive at g(x)? Be specific.
e. How does the completed square form relate to the quadratic parent function, f(x) = x2?
Lesson 21 Summary
Transformations of the quadratic parent function,f(x) = x2, can be rewritten in form g(x) = a(x - h)2 + k where (h, k) is the vertex of the translated and scaled graph of f, with the scale factor of a, the leading coefficient.
We can then quickly and efficiently (without the use of technology) sketch the graph of any quadratic function in the form f(x) = a(x - h)2 + k, using transformations of the graph of the quadratic parent function, f(x) = x2
Lesson 21 Problem Set Sample Solutions
1. Write the function g(x) = -2x2 - 20x -53 of the parent function, in completed square form. Describe the transformations of the graph f(x) = x2, that result in the graph of g.
3. Without using a graphing calculator, sketch the graphs of the functions below based on transformations of the graph of the parent function f(x) = x2. Use your own graph paper and label your graphs.
Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations.