Question

4. What key features of a quadratic graph can be identified and how are the graphs affected when the constants or coefficients are added to the parent quadratic equations? Compare the translations to the graph of linear function. Create examples of your own to explain the differences and similarities.

Answer 1

standard quadratic equation .. y = a x^2 + b x + c 
parabola 'a' not equal to zero 
a<0 parabola opens downward 
a>0 parabola opens upward 
when |a| >>0 the parabola is narrower 
when |a| is close to zero , the parabola is flatter 

when the constant is varied it only effects the vertical position of the parabola , the shape remains the same

Answer 2

Answer:

The parent function of a quadratic function is

$y=x^{2}$

Now, if we add units we can move this function upwards, downwards leftwards and rightwards.

If we add a positive number to the x-variable, then the graph will move to the left.

If we add a negative number to the x-variable, then the graph will move to the right.

If we add a positive number to y-variable, then the graph will move upwards.

If we add a negative number to y-variable, then the graph will move downwards.

So, if we have the function

$y=(x+3)^{2}-5$

We would now that this functions is a parent function moved 3 units leftwards and 5 units downwards, that's how you deduct the translations.

Now, if we compare the rules we use before with linear function, there's no distinction between horizontal and vertical movements, because if we add to x-variable, then y-variable will be also affected.

For example, consider the parent function $y=x$

If we transform it to $y=x+1$, then the whole line will move up 1 unit, or it will move to the left 1 unit, it's the same. The images attached show this.

The second image attached show the transformation applied to the quadratic function. Notice the difference between the linear function and the quadratic function transformation.