Functions and Graphs: Types, Rules, and Plotting

TAKEAWAYS:

• A function assigns exactly one output to each input (vertical line test).
• Types include polynomial, rational, exponential, logarithmic, and piecewise functions.
• Transformations (shifts, stretches, reflections) change graphs predictably.
• Domain/range define where a function is valid and its output limits.
• Graphing requires plotting key points, asymptotes, and intercepts.
• NEB exams test definitions, transformations, and real-world applications.

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1. What is a Function?

A function is a rule that connects inputs (domain) to outputs (range) such that each input has exactly one output. Think of it like a machine:

• Input → Function Rule → Output.
• Example: If $f(x)=2x+3$, then $f(1)=5$, $f(2)=7$, etc.

Key Terms

• Domain: All possible input values (e.g., $x\ne 0$ for $\frac{1}{x}$).
• Range: All possible output values (e.g., $y>0$ for $y=x^2$).
• Vertical Line Test: If a vertical line crosses a graph more than once, it’s not a function.

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2. Types of Functions

(A) Algebraic Functions

1. Polynomial Functions

   • Form: $f(x)=a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_0$.
   • Example: $f(x)=3x^3-2x+1$ (cubic function).
   • Graph: Smooth curves (no breaks or sharp turns).

2. Rational Functions

   • Form: $f(x)=\frac{P(x)}{Q(x)}$ (ratio of polynomials).
   • Example: $f(x)=\frac{1}{x-2}$.
   • Graph: Has vertical asymptotes (where denominator = 0) and horizontal asymptotes (behavior as $x\to \infty$).

3. Piecewise Functions
   • Defined by different rules for different intervals.
   • Example: $$f(x)=\{\begin{array}{ll}x+1& \text{if }x<0,\\ x^2& \text{if }x\ge 0.\end{array}$$
   • Graph: Looks like "broken" pieces.

(B) Trigonometric Functions

• Sine, Cosine, Tangent: Defined using the unit circle.
  • $\sin \theta =\frac{\text{opposite}}{\text{hypotenuse}}$.
  • $\cos \theta =\frac{\text{adjacent}}{\text{hypotenuse}}$.
  • $\tan \theta =\frac{\sin \theta }{\cos \theta }$.
• Graphs: Repeating waves (periodic) with amplitude and period.

(C) Exponential and Logarithmic Functions

1. Exponential: $f(x)=a^x$ (e.g., $y=2^x$).

   • Graph: Always increasing if $a>1$, always decreasing if $0<a<1$.
   • Asymptote: $y=0$ (x-axis).

2. Logarithmic: $f(x)={\log }_{a}x$ (inverse of exponential).

   • Graph: Defined only for $x>0$, passes through $(1,0)$.
   • Asymptote: $x=0$ (y-axis).

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3. Domain and Range

Example: Find the domain and range of $f(x)=\sqrt{4-x^2}$.

• Domain: $4-x^2\ge 0$ → $x^2\le 4$ → $-2\le x\le 2$.
• Range: $\sqrt{4-x^2}\ge 0$ and max at $x=0$: $f(0)=2$. → $0\le y\le 2$.

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4. Transformations of Functions

Graphs can be shifted, stretched, or reflected using transformations:

Example: Sketch $y=f(x+2)-3$ if $f(x)=x^2$.

1. Start with $y=x^2$ (parabola).
2. Shift left by 2: $y=(x+2{)}^2$.
3. Shift down by 3: $y=(x+2{)}^2-3$.

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5. Graphing Techniques

To graph a function like $y=\frac{x^2-1}{x-1}$:

1. Simplify: $y=\frac{(x-1)(x+1)}{x-1}=x+1$ but $x\ne 1$ (hole at $x=1$).
2. Find intercepts:
   • x-intercept: Set $y=0$ → $x=-1$.
   • y-intercept: Set $x=0$ → $y=1$.
3. Asymptotes:
   • Vertical: $x=1$ (denominator zero).
   • Horizontal: As $x\to \infty$, $y\approx x$ (no horizontal asymptote).
4. Plot points: $(0,1)$, $(2,3)$, $(-2,-1)$.

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6. Applications of Functions

1. Physics: $s(t)=ut+\frac{1}{2}a{t}^2$ (distance vs. time).
2. Economics: $P(x)=100-2x$ (price vs. quantity).
3. Biology: $P(t)=P_0{e}^{rt}$ (population growth).

Example: A ball’s height $h(t)=-5t^2+20t$ (in meters).

• Find height at $t=1$ second: $h(1)=-5(1{)}^2+20(1)=15$ m.
• Find when it hits the ground ($h(t)=0$): $-5t^2+20t=0$ → $t(-5t+20)=0$ → $t=0$ or $t=4$ seconds.

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Exam Tip

1. Definitions: Know what a function, domain, and range are.
2. Graphing: Always find intercepts, asymptotes, and key points.
3. Transformations: Practice shifting/stretching graphs.
4. Piecewise Functions: Check each interval separately.
5. NEB Questions:
   • Sketch graphs of given functions.
   • Find domain/range from equations.
   • Apply transformations to match graphs.
   • Solve real-world problems using functions.

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NEB-Style Questions

Short Answer

1. Define a piecewise function with an example.
2. What is the domain of $f(x)=\sqrt{x-3}$?
3. Sketch $y=|x-2|$ and describe its transformations from $y=|x|$.

Long Answer

4. Given $f(x)=\frac{x^2-4}{x-2}$:

   • Simplify and state the domain.
   • Sketch its graph.
   • Find $f(3)$ and $f(-1)$.

5. A function is defined as: $$f(x)=\{\begin{array}{ll}{x}^2+1& \text{if }x\le 0,\\ 2x-1& \text{if }x>0.\end{array}$$

   • Find $f(-2)$ and $f(3)$.
   • Sketch the graph.
   • Determine if $f(x)$ is continuous at $x=0$.

Graph Matching

6. Match the following functions to their graphs (A, B, C, D):
   • $y=x^3$
   • $y=\sqrt{x}$
   • $y=\frac{1}{x}$
   • $y=-|x|$

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Note: Practice graphing at least 5 functions daily. Use graph paper or online tools like Desmos to visualize transformations!