How does the bx value in a quadratic equation affect the graph?
Struggling to understand how the bx value in the equation ax^2 +bx + c affects the quadratic graph that it forms.
I understand how the c value affects the graph and how the ax^2 affects the graph, but when I change the bx value I'm struggling to understand why it affects the graph the way it does if that makes sense.
Because initially thought it would change the graph's shape until I used a graph generator and saw that it seems to move the graph, but not change the scale or the shape of the graph which goes against what I thought it would do.
Any help would be appreciated.
Math Gnome
1 Answer
You can understand the effect of $b$ by completting the square:
\[ax^2+bx+c=a(x^2+\frac{b}{a}x)+c=a(x^2+\frac{b}{a}x+(\frac{b}{2a})^2-(\frac{b}{2a})^2)+c\]
\[=a((x+\frac{b}{2a})^2-(\frac{b}{2a})^2)+c=a(x+\frac{b}{2a})^2-\frac{b^2}{4a}+c.\]
Thus
\[ax^2+bx+c=a(x^2+\frac{b}{a}x)+c=a(x+\frac{b}{2a})^2-\frac{b^2}{4a}+c,\]
which is a parabola. Note when $a>0$ increasing $b>0$ moves the vertex of the parabola $(-\frac{b}{2a},-\frac{b^2}{4a}+c)$ to the left and down.
When $a<0$, increasing $b>0$ moves the vertex of the parabola $(-\frac{b}{2a},-\frac{b^2}{4a}+c)$ to the right and up.
Erdos
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Let me know if you need any clarifications.
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