# Solver to Analyze and Graph a Quadratic Function

\( \) \( \)\( \)An interactive step by step worksheet that determines the vertex, range, minimum or maximum, the intervals of increase and decrease, the x and y intercepts and axis of symmetry of the graph of a given quadratic function. Finally all calculated parameters are put together to plot the graph of the given quadratic function.

Step by step solution
STEP 1: The given quadratic function is written in the form $f(x)=ax^2+bx +c$, identify the values of coefficients $a$, $b$ and $c$.
STEP 2: Use the formulas to determine the $x$- and $y$-coordinates of the vertex of the graph (called parabola) given respectively by $h=-\dfrac{b}{2a}$ and $k=f(h)$, of the of the given quadratic function. Of course you may use other methods to find $h$ and $k$.
STEP 3: Use the sign of coefficient $a$ and the $y$-coordinate of the vertex to determine the range of values of the given quadratic function.
STEP 4: Use the range to determine the minimum or maximum value of the given function and the intervals of increase and decrease.
STEP 5: The $x$-intercepts are points of intersection of the $x$-axis and the graph of $f$. Any point on the $x$-axis has a $y$-coordinate equal to zero. The $x$-intercepts are therefore found by solving the equation $f(x)=0$.
STEP 6: The $y$-intercepts is the point of intersection of the $y$-axis and the graph of $f$. Any point on the $y$-axis has the $x$-coordinate equal to zero. The $y$-intercept is therefore given by $f(0)$.
STEP 7: The graph of $f$ is a parabola and has an axis of symmetry which a vertical line given by $x=h$.
Below are shown the graph of $f$ as a parabola (green), the x-intercepts (if any) as the intersetion (s) of the parabola and the x-axis (red), the y-intercept as the intersection of the parabola and y-axis (brown), the vertex as a maximum or minimum point (black) and the axis of symmetry as a dashed line (magenta). (Change scales if necessary) |