Lessons 6 · 1 lessons · Sh.A. Alimov, A.R. Xalmukhamedov, M.A. Mirzaakhmedov. Algebra: textbook for Grade 9 of general secondary schools. Revised 4th edition. Tashkent: “O‘qituvchi”, 2019
6
Graphing a quadratic function
Textbook: pp. 18–23
GoalGraph a quadratic function by a plan and read its properties from the graph; find the greatest or smallest value.
New words
vertex of a parabola · parabola uchizeros of a function · funksiyaning nollariintervals of increase and decrease · o‘sish va kamayish oraliqlarigreatest (smallest) value · eng katta (eng kichik) qiymat
Explanation
The graph of y = ax² + bx + c is drawn by this plan. 1) Vertex: find x₀ = −b : (2a) and y₀ = y(x₀) and mark the point (x₀, y₀). 2) Draw the axis of symmetry through the vertex, parallel to Oy. 3) Solve ax² + bx + c = 0 to find the zeros (if any). 4) Take two points symmetric about the axis, for example x = 0 and x = 2x₀ (both give y = c). 5) Join the points by a smooth curve. From the graph we read the sign of the function and its intervals of increase and decrease. If a > 0 the function has its smallest value y₀ at x₀, and if a < 0 its greatest value y₀. So problems such as “greatest area” or “greatest product” reduce to finding the vertex of a quadratic function.
Worked examples
y = x² − 2x − 3: vertex (1, −4); zeros x = −1 and x = 3; y(0) = y(2) = −3. From the graph: smallest value −4 (at x = 1); y < 0 for −1 < x < 3, y > 0 for x < −1 or x > 3; the function decreases for x ≤ 1 and increases for x ≥ 1.
A rectangle has perimeter 20 m; if one side is x m, the other is 10 − x. Its area is S = x(10 − x) = −x² + 10x. Here a = −1 < 0, x₀ = 5, S(5) = 25. The greatest area is 25 m², a square with side 5 m.
Class activity
“Graph workshop”: groups draw the same function (for example y = x² − 6x + 8) in five steps; a different student does each step, and at the end the groups compare results.
Practice
1
Find the vertex and zeros of y = x² − 6x + 8.
vertex (3, −1); zeros x = 2 and x = 4
2
Find the greatest value of y = −x² + 2x + 3.
4
3
Does the graph of y = x² + 2x + 3 meet the Ox axis?
No: D = 4 − 12 = −8 < 0; the parabola lies entirely above Ox (smallest value 2).
4
Two numbers have sum 12. For the product to be greatest, what are the numbers and what is the product?