☰ Contents · Mathematics

Trigonometric inequalities and graph transformations

Lessons 31–32 · 2 lessons · M. A. Mirzaahmedov, Sh. N. Ismailov, A. Q. Amanov (algebra and analysis), B. Q. Haydarov (geometry). Mathematics Grade 10, Parts 1 and 2, 1st edition. EXTREMUM PRESS, Tashkent, 2017
32

Transformations of graphs

Textbook, Part 2: pp. 48–50
GoalObtain graphs of functions y = m·f(k(x − a)) + b from the graph of y = f(x) by shifts, stretches and reflections.
New words
translation · parallel ko‘chirishstretch · cho‘zishcompression · siqishreflection · simmetriya
Explanation

From the graph of y = f(x), the graph of y = f(x − a) + b is obtained by translating by the vector (a, b): a units right (left if a < 0) and b units up (down if b < 0). The graph of y = −f(x) is the reflection of y = f(x) in the Ox axis. In y = m·f(x) the ordinates are stretched |m| times (|m| > 1) or compressed (|m| < 1). In y = f(kx) the abscissas are compressed |k| times (|k| > 1) or stretched (|k| < 1). When several transformations are applied one after another, the order matters: for example y = 2 − (x + 3)² is obtained from y = x² by shifting 3 units left, reflecting in Ox, and then raising by 2 units. The graph of y = sin 2x comes from y = sin x by compressing it twice towards the Oy axis, so its period is 2π/2 = π.

Worked examples
y = (x − 2)² + 1: the graph of y = x² is shifted 2 units right and 1 unit up; the vertex goes from (0, 0) to (2, 1).
Let f(1) = 4. For y = 3f(x − 1) + 2 at x = 2 we get y = 3f(1) + 2 = 3 · 4 + 2 = 14: the point (1, 4) goes to (2, 14) (1 right, ordinate stretched 3 times, 2 up).
Class activity

On graph paper draw y = x², copy it onto tracing paper and slide it to obtain y = (x − 2)² + 1; then check both graphs point by point.

Practice
1
How is the graph of y = f(x − 3) obtained from y = f(x)?
2
y = (x + 2)² − 5 is obtained from y = x². Where is the vertex?
3
If f(2) = 5, find the value of y = 2f(x + 1) − 3 at x = 1.
4
Why does the graph of y = f(x − a) with a > 0 shift to the right, despite the minus sign?