The quadratic function and properties of functions
Find the vertex of a parabola by the formula, read the properties of a function from its graph, solve a quadratic inequality graphically and draw the graphs of y = |f(x)| and y = f(|x|).
- Germany: grade 9
- Ukraine: grade 9
- Russia: grade 9
Guide
In Ukrainian Grade 9 algebra (§8, §11, §12) the graph of a function shows its properties: the zeros, the intervals where it keeps its sign, where it increases or decreases, the domain D(y) and the range E(y).
A quadratic function is y = ax2 + bx + c with a ≠ 0. Completing the square gives y = a(x − x0)2 + y0, where x0 = −b2a and y0 is the value of the function at x0. The vertex of the parabola is the point (x0,y0).
The graphical method for f(x) > 0: sketch the parabola from its zeros and the sign of a, then find where the graph is above the x-axis. If there are several intervals, join them with the union sign ∪.
To draw y = |f(x)|, flip the part of y = f(x) that lies below the x-axis up in the axis. To draw y = f(|x|), keep the part to the right of the y-axis and mirror it in the y-axis.