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|).
- أوكرانيا: الصف ٩
- روسيا: الصف ٩
- ألمانيا: الصف ٩
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الدليل
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 = ax٢ + bx + c with a ≠ ٠. Completing the square gives y = a(x − x٠)٢ + y٠, where x٠ = −b٢a and y٠ is the value of the function at x٠. The vertex of the parabola is the point (x٠,y٠).
The graphical method for f(x) > ٠: 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.