Every parabola represented by the equation $y = ax^2 + bx + c$ can be obtained by stretching and translating the graph of $y = x^2$.
Therefore:
The sign of the leading coefficient, $-a$ or $a$, determines if the parabola opens up or down i.e.
The leading coefficient, $a$, also determines the amount of vertical stretch or compression of $y = x^2$ i.e.
The constant term, $c$, determines the vertical translation of $y = x^2$ i.e.
Now for $bx$. Initially, I thought it would determine the amount of horizontal translation since the constant term, $c$, already accounted for the vertical translation, but when I plugged in some quadratics the graph of $y = x^2$ translated both horizontally and vertically. Here are the graphs:
Seeing as the middle term, $bx$, does more than just horizontally translate, how do you describe its effect on $y=x^2$? Would it be accurate to say that it both horizontally and vertically translates the graph of $y = x^2$?
Best Answer
Yes, it will effect both a horizontal and vertical translation, and you can see how much by completing the square. For example, $$x^2+3x=\left(x+\frac32\right)^2-\frac94$$
Compare that to your graph of $y=x^2+3x$. Of course, if the coefficient of the quadratic term is not $1$ things get a little more complicated, but you can always see what the graph the graph will look like by completing the square.