Are all quadratics of the form y = ax^2+bx +c and y = -ax^2+bx +c functions? Explain your answer.
No, all quadratics in x and y are of the form
ax^2 + bxy + cy^2 + dx + ey + f = 0
Now, if you want quadratics where y is expressed as a function of x, then all are of the form
y = ax^2 + bx + c
where a,b,c can be positive or negative. There's no need for -ax^2 + bx + c.
Yes, both y = ax^2+bx+c and y = -ax^2+bx+c are examples of quadratic functions.
To determine if a given equation represents a function, we need to check whether each input (x-value) corresponds to a unique output (y-value). In other words, for every x-value we plug into the equation, we should obtain only one y-value as a result.
In the case of quadratic functions of the form y = ax^2+bx+c and y = -ax^2+bx+c, we can see that the term with the highest power of x is x^2. Since x^2 is never negative or positive zero, it does not change the sign of the quadratic expression. So, whether there is a positive or negative a coefficient in front of x^2 does not affect the fact that for each input x, we will get a unique output y. Therefore, both equations y = ax^2+bx+c and y = -ax^2+bx+c represent functions.
In conclusion, any quadratic equation in the form y = ax^2+bx+c or y = -ax^2+bx+c is a function because it satisfies the criteria of producing a unique output (y-value) for every input (x-value).