Tier 6: Functions & Graphs

🧠 What Is a Function?

A function assigns each input exactly one output. Written as \( f(x) \).

✏️ Example 1

If \( f(x) = 2x + 3 \), then \( f(4) = 2(4) + 3 = 11 \)

✏️ Example 2

If \( g(x) = x^2 - 1 \), then \( g(-2) = (-2)^2 - 1 = 3 \)

✏️ Example 3

If \( h(x) = \sqrt{x} \), then \( h(16) = 4 \)

✏️ Example 4

If \( f(x) = \frac{1}{x} \), find \( f(2) \) and \( f(-1) \)
\( f(2) = 0.5 \), \( f(-1) = -1 \)

🧠 Linear Function Form

Standard form is \( y = mx + b \) where \( m \) is slope and \( b \) is y-intercept.

✏️ Example 1

\( y = 2x + 3 \): Slope = 2, y-intercept = 3

✏️ Example 2

\( y = -x + 4 \): Slope = -1, y-intercept = 4

✏️ Example 3

\( y = \frac{1}{2}x - 2 \): Slope = 1/2, y-intercept = -2

✏️ Example 4

\( y = -3x \): Slope = -3, y-intercept = 0

✏️ Example 1

A line goes through (0, 2) and (1, 4). Slope = \( \frac{4 - 2}{1 - 0} = 2 \)

✏️ Example 2

A graph with constant y = 5 means slope is 0.

✏️ Example 3

A negative slope means the line is decreasing.

✏️ Example 4

Distance vs. time graph: horizontal line means object is stopped.

📊 Function Graphs

Observe how each curve behaves—linear grows steadily, quadratic curves, exponential rises quickly, and logarithmic grows slowly.

🧠 Definitions

• \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \)
• \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \)
• \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \)
• \( \csc(\theta) = \frac{1}{\sin(\theta)} \)
• \( \sec(\theta) = \frac{1}{\cos(\theta)} \)
• \( \cot(\theta) = \frac{1}{\tan(\theta)} \)

📐 Common Angle Values

• \( \sin(\frac{\pi}{6}) = \frac{1}{2}, \cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2}, \tan(\frac{\pi}{6}) = \frac{1}{\sqrt{3}} \)
• \( \sin(\frac{\pi}{4}) = \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}, \tan(\frac{\pi}{4}) = 1 \)
• \( \sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}, \cos(\frac{\pi}{3}) = \frac{1}{2}, \tan(\frac{\pi}{3}) = \sqrt{3} \)
• \( \sin(0) = 0, \cos(0) = 1, \tan(0) = 0 \)
• \( \sin(\pi) = 0, \cos(\pi) = -1, \tan(\pi) = 0 \)

✏️ Examples

1. \( \sin(\frac{\pi}{2}) = 1 \)
2. \( \cos(\pi) = -1 \)
3. \( \tan(\frac{\pi}{4}) = 1 \)
4. \( \csc(\frac{\pi}{6}) = \frac{1}{\sin(\frac{\pi}{6})} = 2 \)
5. \( \sec(\frac{\pi}{3}) = \frac{1}{\cos(\frac{\pi}{3})} = 2 \)
6. \( \cot(\frac{\pi}{3}) = \frac{1}{\sqrt{3}} \)
7. \( \sin(\pi) = 0 \Rightarrow \csc(\pi) \text{ undefined} \)
8. \( \tan(0) = 0 \Rightarrow \cot(0) \text{ undefined} \)
9. \( \sec(\frac{\pi}{2}) = \frac{1}{0} \Rightarrow \text{undefined} \)
10. \( \tan(\pi) = 0 \Rightarrow \cot(\pi) \text{ undefined} \)

📗 Identity Reminder

• \( \sin^2(x) + \cos^2(x) = 1 \)
• \( 1 + \tan^2(x) = \sec^2(x) \)
• \( 1 + \cot^2(x) = \csc^2(x) \)

🧠 Key Concepts

• Use identities: \( \sin^2(x) + \cos^2(x) = 1 \)
• Reciprocal identities: \( \csc(x) = \frac{1}{\sin(x)} \), etc.
• Quotient identities: \( \tan(x) = \frac{\sin(x)}{\cos(x)} \), \( \cot(x) = \frac{\cos(x)}{\sin(x)} \)

✅ You Can:

• Simplify: \( \frac{\sin(x)}{\cos(x)} \rightarrow \tan(x) \)
• Convert \( \frac{1}{\csc(x)} \rightarrow \sin(x) \)
• Factor: \( \sin^2(x)(1 + \cot^2(x)) \rightarrow \sin^2(x)\csc^2(x) = 1 \)

⛔ Avoid Mistakes:

• \( \sin(x + y) \ne \sin(x) + \sin(y) \)
• \( \frac{1}{\sin(x) + \cos(x)} \ne \csc(x) + \sec(x) \)
• \( \tan^2(x) \ne \tan(x^2) \)

✏️ Examples

1. Simplify: \( \frac{\sin(x)}{\cos(x)} = \tan(x) \)
2. Simplify: \( 1 - \sin^2(x) = \cos^2(x) \)
3. Simplify: \( \frac{1}{\sec(x)} = \cos(x) \)
4. Simplify: \( \frac{\sin(x) \cdot \cot(x)}{\csc(x)} \rightarrow \cos(x) \)
5. Simplify: \( \csc(x) - \cot(x) = \frac{1 - \cos(x)}{\sin(x)} \)
6. Simplify: \( \sec^2(x) - \tan^2(x) = 1 \)
7. Combine: \( \frac{1}{\cos(x)} + \tan(x) \rightarrow \sec(x) + \tan(x) \)
8. Simplify: \( \frac{\sin^2(x)}{1 + \cos(x)} + \frac{1 - \cos(x)}{1 + \cos(x)} \rightarrow 1 \)
9. Factor: \( \sin^2(x) - \cos^2(x) = -\cos(2x) \)
10. Simplify: \( \cot(x) \cdot \tan(x) = 1 \)

✅ Practice Tips

• Always check domains—avoid dividing by zero.
• Use common identities to rewrite everything in terms of sine and cosine.

📈 Trigonometric Graphs

The sine and cosine graphs oscillate smoothly, while the tangent graph has asymptotes where \( \cos(x) = 0 \).