Exercise 26 Page 159

We want to describe how to transform the parent function f(x)=x^2 to the graph of the given quadratic function. f(x)=4x^2-18 To do so, we need to consider two possible transformations.

1. Stretches and compressions
2. Vertical translations

   Let's consider them one at the time.

   Stretch or Compression

   We have a vertical stretch when x^2 is multiplied by a number whose absolute value is greater than one. If x^2 is multiplied by a number whose absolute value is less than one, a vertical compression will take place.
   

   If x^2 is being multiplied by a negative number, the above still applies but everything will be upside down. In the given exercise, x^2 is multiplied by 4. Therefore, the previous graph will be vertically stretched by a factor of 4.

   

   Vertical Translation

   If an addition or subtraction is applied to the whole function, the graph will be vertically translated. In the case of addition, the graph will be translated up. In the case of subtraction, it will be moved downwards. In the given equation, 18 is subtracted to the whole function, so the previous graph will be translated eighteen units down.

   

   Final Graph

   Let's now graph the given function and the parent function f(x)=x^2 on the same coordinate grid.

   

   Finally, let's summarize how to draw the graph of the given function when starting with the parent function, f(x)=x^2.

   • Vertical stretch by a scale factor of 4.
   • Translation eighteen units down.