We will be comparing Equation One, y_1=ax^2, and Equation Two, y_2=a^2x^2.
Equation One:& y_1=ax^2
Equation Two:& y_2=a^2x^2
The difference between the equations is that in Equation Two, it is a^2 instead of just a. Recall that when we square a number, the result is always positive. Therefore, Equation Two will be always be positive, regardless of the a-value, because we are multiplying two positive numbers.
Ifa is positive or negative:
y_2=a^2x^2 ⇒ y_2=(+)(+) ⇒ y_2=(+)
In Equation One, if a is positive we will multiply two positive numbers together, resulting in the function being positive.
Ifa is positive:
y_1=ax^2 ⇒ y_1=(+)(+) ⇒ y_1=(+)
However, when a is negative we will be multiplying a negative value by the always-positive x^2 value. Therefore, when a is negative, the function will be negative.
Ifa is negative:
y_1=ax^2 ⇒ y_1=(-)(+) ⇒ y_1=(-)
In conclusion, this all tells us that when a is positive, both Equation One and Two will be positive. However, when a is negative Equation One will be negative and Equation Two will be positive. Thus, the graphs will be in the same quadrant when a is positive.
b
We want to see which parabola will be wider. Recall that wider parabolas have smaller |a|-values. Since we want to know when y_1=ax^2 will be wider than y_2=a^2x^2, we want to find when |a| will be smaller than |a^2|. Note that a^2 is always positive, so we do not need absolute value signs around it.
|a|
For every |a|-value greater than one, we can see that |a|
|a|
a^2
2
4
3
9
4
16
We can see that when |a| is greater than one, |a|less than or equal to one and greater than zero. That is, 0<|a|≤ 1. Let's test some values to see what happens.
|a|
a^2
1
1^2=1
1/2
1^2/2^2=1/4
1/4
1^2/4^2=1/16
We see that when 0<|a|≤ 1, |a| is not less than than a^2. Therefore, this will not be part of our solution.
|a|1
In summation, we have found that |a|1. This means that the graph of y_1=ax^2 will be wider than y_2=a^2x^2 when |a|>1.
