Exercise 29 Page 424

We want to determine if the graph of f(x)= ax^2 is wider than the graph of g(x)= dx^2 when | a|>| d|. To do so, we will use two facts.

1. The graph of f(x)= ax^2 is narrower than the graph of h(x)=x^2 when | a|>1.
2. The graph of f(x)= ax^2 is wider than the graph of h(x)=x^2 when 0< | a|<1.

We will examine the graphs in three cases.

When |a|>|d|>1

As a result of the first fact, the greater the a-value, the narrower the graph of a function of the form f(x)=ax^2. Hence, the graph of f will be narrower than the graph of g when | a|>| d|>1.

When |a|>1>|d|

Combining two facts, the graph of the function f is apparently narrower than the graph of g since | a| is greater than 1, and | d| is less than 1.

When 1>|a|>|d|

As a result of the second fact, the smaller the a-value, the wider the graph of a function of the form f(x)=ax^2. Hence, the graph of g will be wider than the graph of f when 1>| a|>| d|.

We see that in all cases the graph of f is narrower than the graph of g. The cases above are also true for the reflection in the x-axis of the graphs of the functions. Hence, the graph of the function f is never wider than the graph of g.