Question

what does the coefficient ( a ) in the standard form ( y = ax^2 + bx + c ) primarily affect?

a. the y - intercept
b. the axis of symmetry
c. the x - intercepts
d. the direction and width of the parabola

Explanation:

In the quadratic standard form $y=ax^2+bx+c$:

• The y-intercept is determined by $c$ (when $x=0$, $y=c$).
• The axis of symmetry is given by $x=-\frac{b}{2a}$, which depends more on the relationship between $a$ and $b$, not $a$ alone.
• The x-intercepts are found by solving $ax^2+bx+c=0$, which depends on all coefficients.
• The coefficient $a$: if $a>0$, the parabola opens upward; if $a<0$, it opens downward. The absolute value $|a|$ determines width: larger $|a|$ makes the parabola narrower, smaller $|a|$ makes it wider.

Answer:

d. The direction and width of the parabola