Use this free online calculator to find the vertex, x-intercepts, and y-intercept of a quadratic function, and automatically rewrite it in vertex form.
The quadratic function is assumed to be:
\[ f(x)=ax^2+bx+c \]The vertex is the maximum or minimum point of a parabola. This calculator uses:
\[ h=-\frac{b}{2a} \] \[ k=f(h) \]to compute the vertex coordinates \( (h,k) \).
Once \(h\) and \(k\) are known, the vertex form becomes:
\[ f(x)=a(x-h)^2+k \]To find x-intercepts, the calculator solves:
\[ ax^2+bx+c=0 \] using the quadratic formula: \[ x_1=\frac{-b+\sqrt{\Delta}}{2a}, \quad x_2=\frac{-b-\sqrt{\Delta}}{2a} \] where \[ \Delta=b^2-4ac \]The y-intercept is simply:
\[ f(0)=c \]Enter coefficients \(a\), \(b\), and \(c\), choose decimal precision, then click Solve. The calculator displays:
If the x-intercepts are equal, the parabola touches the x-axis at exactly one point.