☰ Contents · Algebra

The quadratic function and y = x²

Lessons 2–3 · 2 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
3

The function y = x²

Textbook: pp. 7–9
GoalKnow the graph of y = x² (a parabola) and its properties; find intersection points of a parabola and a line.
New words
parabola · parabolavertex of a parabola · parabolaning uchiaxis of symmetry · simmetriya o‘qiincreasing and decreasing function · o‘suvchi va kamayuvchi funksiya
Explanation

The graph of y = x² is drawn by making a table of values, plotting the points and joining them with a smooth curve; this curve is called a parabola. For x ≠ 0 we have y > 0 and for x = 0 we have y = 0, so the parabola passes through the origin and all its other points lie above the Ox axis. Because (−x)² = x², the graph is symmetric about the Oy axis; the point where a parabola meets its axis of symmetry is called its vertex, here the vertex is (0, 0). The function is increasing for x > 0 and decreasing for x < 0. To find the intersection points of a parabola and a line we make a system from the equations of both functions.

Worked examples
The point A(−4, 16) lies on y = x², because (−4)² = 16. B(3, 6) does not: 3² = 9 ≠ 6.
Intersection of y = x² and y = 2x + 3: x² = 2x + 3, x² − 2x − 3 = 0, x₁ = 3, x₂ = −1. y₁ = 9, y₂ = 1. Answer: (3, 9), (−1, 1).
Class activity

“Draw the parabola”: each student marks the points for x = −3, −2, ..., 3 on squared paper; in pairs they check each other's points and join them with a smooth curve.

Practice
1
Does C(−5, 25) lie on the parabola y = x²?
2
Find the intersection points of the parabola y = x² and the line y = x + 2.
3
Compare y(−3) and y(−2) for y = x².
4
Why is the parabola y = x² symmetric about the Oy axis?