☰ Contents · Algebra

The function y = ax²

Lessons 4 · 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
4

The function y = ax²

Textbook: pp. 10–13
GoalObtain the graph of y = ax² from the graph of y = x² and know its properties.
New words
branches of a parabola · parabola tarmoqlaristretching · cho‘zishcompression · siqishgreatest (smallest) value · eng katta (eng kichik) qiymat
Explanation

The graph of y = ax² (a ≠ 0) is also a parabola with its vertex at the origin. We can obtain it from the graph of y = x²: the ordinate of every point is multiplied by a. If a > 1 the graph is stretched along the Oy axis (the parabola becomes narrower), and if 0 < a < 1 it is compressed (the parabola becomes wider). If a < 0 the graph is the mirror image, in the Ox axis, of the graph of y = |a|x². For a > 0 the branches point up, for a < 0 they point down. Properties: the graph is symmetric about the Oy axis; for a > 0 the function increases for x > 0, decreases for x < 0 and has the smallest value 0 at x = 0; for a < 0 it decreases for x > 0, increases for x < 0 and has the greatest value 0 at x = 0.

Worked examples
Graph of y = 3x²: at x = 1, y = 3 and at x = −2, y = 12, three times higher than on y = x². For y = −2x², x = 1 gives y = −2 and x = −3 gives y = −18: the parabola opens downward.
Intersection of y = 2x² and y = 3x − 1: 2x² = 3x − 1, 2x² − 3x + 1 = 0, D = 9 − 8 = 1, x = (3 ± 1) : 4, so x₁ = 1, x₂ = 1/2. The points are (1, 2) and (1/2, 1/2).
Class activity

“Parabola family”: students draw y = x², y = 2x², y = x²/2 and y = −x² in different colours on one coordinate plane and decide which parabola is the narrowest.

Practice
1
Which way do the branches of y = −x²/2 point?
2
Find the value of y = 3x² at x = −2.
3
Does A(2, −8) lie on the parabola y = −2x²?
4
Why is the greatest value of y = ax² equal to 0 when a < 0?