The unit begins by contrasting linear growth with quadratic curvature. The quadratic parent function is defined as $f(x) = x^2$. Its graph forms a smooth, symmetric U-shaped curve known as a parabola. Unlike linear graphs that maintain a constant slope, the slope of a parabola changes at every single coordinate point.
When rewritten into the standard form of quadratic equation, the formula expands to $f(x) = ax^2 + bx + c$. Each coefficient directly dictates how the curve behaves on the coordinate plane:
The leading coefficient $a$ controls the direction of opening and vertical stretch or compression. If $a > 0$, the parabola opens upward, resembling a bowl that holds water. If $a < 0$, the parabola flips upside down and opens downward. If the absolute value $|a|$ is greater than 1, the curve narrows sharply toward the vertical axis; if $|a|$ falls between 0 and 1, the curve flattens outward.
The constant term $c$ represents the vertical offset. Finding y-intercepts requires evaluating the function at $x = 0$. Because $a(0)^2 + b(0) = 0$, the graph crosses the vertical axis precisely at $(0, c)$. When an Edgenuity problem asks for the vertical intercept of $y = 3x^2 - 5x + 7$, no calculation is necessary: the coordinate is simply $(0, 7)$.
