y = −f(x) and y = f(−x), the graphs of sin x, cos x and tan x, exact values, and solving sin x = k and cos x = k.
What you will learn
- Reflect graphs and points: y = −f(x) and y = f(−x)
- Recall exact values of sin, cos and tan
- Solve sin x = k and cos x = k for 0° to 360° using the graphs
Two reflections
- y = −f(x) changes the sign of every y value: a reflection in the x-axis. (3, 5) goes to (3, −5).
- y = f(−x) changes the sign of every x value: a reflection in the y-axis. (3, 5) goes to (−3, 5).
For f(x) = x² + 2x − 3: −f(x) = −x² − 2x + 3, and f(−x) = (−x)² + 2(−x) − 3 = x² − 2x − 3. In f(−x), even powers of x keep their sign and odd powers change sign.
Reflections and translations combine: do them one at a time. If y = f(x) has a maximum at (4, 3), y = f(−x) has it at (−4, 3), and y = f(−x) + 2 at (−4, 5). The same works for trig graphs: y = −sin x turns the maximum at (90°, 1) into a minimum at (90°, −1), and y = 2 − sin x moves that minimum up to (90°, 1).
The rest of this lesson and its 7 practice questions are locked.
In this lesson
- Two reflections
- The graphs of sin x, cos x and tan x
- Two solutions from one
- Try it: Read the values off the circle
7 practice questions, marked as you go; about 25 minutes in all.
Key terms
X-axis, Y-axis
In this unit: Functions and graph transformations
Function notation, composite and inverse functions, the shapes of standard graphs, translations and reflections of y = f(x), and the graphs of sin, cos and tan (Higher).
- Function notation
- Composite and inverse functions
- Recognising graphs
- Translating graphs
- Reflections and trigonometric graphs