The equation \( y = ax^2 + bx + c \) represents a quadratic function in the variable \( x \).
Here, \( a \), \( b \), and \( c \) are constants, where:
- \( a \): The coefficient of \( x^2 \). This determines the opening direction and the "width" of the parabola. If \( a > 0 \), the parabola opens upwards; if \( a < 0 \), it opens downwards.
- \( b \): The coefficient of \( x \). It affects the position of the vertex and the axis of symmetry of the parabola.
- \( c \): The constant term. It represents the y-intercept of the quadratic function, which is the point where the graph intersects the y-axis (when \( x = 0 \)).
The graph of this quadratic function is a parabola. The vertex of the parabola can be found using the formula:
\[ x = -\frac{b}{2a} \]
The value of \( y \) at this point can be found by substituting \( x \) back into the equation. The quadratic function can also be factored or completed to find roots (solutions for \( y = 0 \)) depending on the discriminant \( D = b^2 - 4ac \):
- If \( D > 0 \): Two distinct real roots.
- If \( D = 0 \): One real root (the vertex is on the x-axis).
- If \( D < 0 \): No real roots (the parabola does not intersect the x-axis).
In summary, the quadratic function \( y = ax^2 + bx + c \) describes a parabolic curve, and its characteristics depend on the values of coefficients \( a \), \( b \), and \( c \).