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
2
Definition of a quadratic function
Textbook: pp. 5–6
GoalKnow the definition of a quadratic function and find its values and zeros.
New words
quadratic function · kvadrat funksiyacoefficient · koeffitsiyentzeros of a function · funksiyaning nollarivariable · o‘zgaruvchi
Explanation
A function of the form y = ax² + bx + c is called a quadratic function, where a, b, c are given real numbers, a ≠ 0, and x is a real variable. If a = 0 the function becomes the linear function y = bx + c, so the condition a ≠ 0 matters. b and c may be zero: y = x² and y = −3x² are quadratic functions too. To find the value of the function we substitute a number for x. The values of x for which the function equals zero are called the zeros of the function; to find them we solve the equation ax² + bx + c = 0. Quadratic functions occur in physics (the height of a thrown object) and in geometry (the area of a square).
Worked examples
For y = x² − 3x − 4: y(−1) = 1 + 3 − 4 = 0, y(2) = 4 − 6 − 4 = −6, y(5) = 25 − 15 − 4 = 6. So x = −1 is a zero of the function.
For which x does y = x² + 2x − 3 equal 5? x² + 2x − 3 = 5, i.e. x² + 2x − 8 = 0, so x₁ = 2, x₂ = −4. Check: 4 + 4 − 3 = 5 and 16 − 8 − 3 = 5.
Class activity
“Quadratic or not?”: the teacher writes formulas y = ... on the board, students raise a “yes” or “no” card and must also name a, b and c.
Practice
1
Which of y = 5x² − 1, y = 7x + 2, y = −x², y = 3x³ + x are quadratic functions?
y = 5x² − 1 and y = −x²
2
Find the zeros of y = x² − 6x + 5.
x₁ = 1, x₂ = 5
3
Find the value of y = x² − 7x + 9 at x = 6.
3
4
Why must a ≠ 0 in a quadratic function?
If a = 0 the term ax² disappears and the function is just the linear function y = bx + c.