Quadratic Questions

 

Let’s observe how the parabola y=ax2 changes as the value of a is varied.

 

First, we graph y=x2, or y=(1)x2.

 

 

This is the simplest parabola to graph. We see that the y and x intercepts are both the origin. Since the function is quadratic, there will be no negative y values. Therefore, f(x) and f(-x) both equal the same y value.

 

Now, let’s observe what happens as we vary the value of a.

 


 

a=0

We see that when a=0 y=(0) x2, or y=0, which is the x-axis. No matter what our input vale for x is, the output value will be 0.

 


a>1

We see that when our a value is greater than 1 the parabola appears to be “stretched”. This is because for every input value of x, the output value is being multiplied by the value of a. Since a is greater than 1 in this case, the output value will be larger than the output value of y=x2. Therefore, when a is greater than 1, the function is increasing at a faster rate than y=x2.

 


0<a<1

We see that when our a value is greater than 0 and less than 1 the parabola appears to be “shrunken”. This is because for every input value of x, the output value is being multiplied by the value of a. Since a is between 0 and 1 in this case, the output value will be smaller than the output value of y=x2. Therefore, when a is between 0 and 1, the function is increasing at a slower rate than y=x2.


 

Negative a

We see that when our a value is negative the parabola is reflected over the x-axis. It does not matter whether the input value is positive or negative. It will become positive when it is squared, then become negative when it is multiplied by a negative value of a. Therefore, when a is negative, all of the output values of the function are negative, as well. We see that the same properties as above hold as well. If 0>a>-1, the parabola appears to be shrunken, and if a<-1 the parabola appears to be stretched.

 


Conclusion

In conclusion, we see that the value of a can greatly affect even the most basic quadratic functions. It can affect the rate at which the function increases or decreases, and the orientation of the parabola.

 

 

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