10 Basic Parent Functions and Transformation Rules

Basic Linear and Quadratic Parent Functions

• Linear Function

  • Equation: $y = x$
  • Classification: Linear, Odd
  • Domain: $(-\infty, \infty)$
  • Range: $(-\infty, \infty)$
  • End Behavior:
    • As $x \rightarrow -\infty$, $y \rightarrow -\infty$
    • As $x \rightarrow \infty$, $y \rightarrow \infty$
  • Critical Points: $(-1, -1)$, $(0, 0)$, $(1, 1)$

• Absolute Value Function

  • Equation: $y = |x|$
  • Classification: Absolute Value, Even
  • Domain: $(-\infty, \infty)$
  • Range: $[0, \infty)$
  • End Behavior:
    • As $x \rightarrow -\infty$, $y \rightarrow \infty$
    • As $x \rightarrow \infty$, $y \rightarrow \infty$
  • Critical Points: $(-1, 1)$, $(0, 0)$, $(1, 1)$

• Quadratic Function

  • Equation: $y = x^2$
  • Classification: Quadratic, Even
  • Domain: $(-\infty, \infty)$
  • Range: $[0, \infty)$
  • End Behavior:
    • As $x \rightarrow -\infty$, $y \rightarrow \infty$
    • As $x \rightarrow \infty$, $y \rightarrow \infty$
  • Critical Points: $(-1, 1)$, $(0, 0)$, $(1, 1)$

Polynomial and Radical Parent Functions

• Square Root Function

  • Equation: $y = \sqrt{x}$
  • Classification: Radical, Neither
  • Domain: $[0, \infty)$
  • Range: $[0, \infty)$
  • End Behavior:
    • As $x \rightarrow 0$, $y \rightarrow 0$
    • As $x \rightarrow \infty$, $y \rightarrow \infty$
  • Critical Points: $(0, 0)$, $(1, 1)$, $(4, 2)$

• Cubic Function

  • Equation: $y = x^3$
  • Classification: Cubic, Odd
  • Domain: $(-\infty, \infty)$
  • Range: $(-\infty, \infty)$
  • End Behavior:
    • As $x \rightarrow -\infty$, $y \rightarrow -\infty$
    • As $x \rightarrow \infty$, $y \rightarrow \infty$
  • Critical Points: $(-1, -1)$, $(0, 0)$, $(1, 1)$

• Cube Root Function

  • Equation: $y = \sqrt[3]{x}$
  • Classification: Cube Root, Odd
  • Domain: $(-\infty, \infty)$
  • Range: $(-\infty, \infty)$
  • End Behavior:
    • As $x \rightarrow -\infty$, $y \rightarrow -\infty$
    • As $x \rightarrow \infty$, $y \rightarrow \infty$
  • Critical Points: $(-1, -1)$, $(0, 0)$, $(1, 1)$

Transcendental and Rational Parent Functions

• Exponential Function

  • Equation: $y = b^x$
  • Conditions: $b > 1$ (Example provided: $y = 2^x$)
  • Classification: Exponential, Neither
  • Domain: $(-\infty, \infty)$
  • Range: $(0, \infty)$
  • End Behavior:
    • As $x \rightarrow -\infty$, $y \rightarrow 0$
    • As $x \rightarrow \infty$, $y \rightarrow \infty$
  • Critical Points: $(-1, \frac{1}{b})$, $(0, 1)$, $(1, b)$
  • Asymptote: Horizontal line at $y = 0$

• Logarithmic Function

  • Equation: $y = \log_b(x)$
  • Conditions: $b > 1$ (Example provided: $y = \log_2(x)$
  • Classification: Log, Neither
  • Domain: $(0, \infty)$
  • Range: $(-\infty, \infty)$
  • End Behavior:
    • As $x \rightarrow 0^{+}$, $y \rightarrow -\infty$
    • As $x \rightarrow \infty$, $y \rightarrow \infty$
  • Critical Points: $(\frac{1}{b}, -1)$, $(1, 0)$, $(b, 1)$
  • Asymptote: Vertical line at $x = 0$

• Rational Function (Inverse)

  • Equation: $y = \frac{1}{x}$
  • Classification: Rational, Odd
  • Domain: $(-\infty, 0) \cup (0, \infty)$
  • Range: $(-\infty, 0) \cup (0, \infty)$
  • End Behavior:
    • As $x \rightarrow -\infty$, $y \rightarrow 0$
    • As $x \rightarrow \infty$, $y \rightarrow 0$
  • Critical Points: $(-1, -1)$, $(1, 1)$
  • Asymptotes: Horizontal line at $y = 0$, Vertical line at $x = 0$

• Rational Function (Inverse Squared)

  • Equation: $y = \frac{1}{x^2}$
  • Classification: Rational, Even
  • Domain: $(-\infty, 0) \cup (0, \infty)$
  • Range: $(0, \infty)$
  • End Behavior:
    • As $x \rightarrow -\infty$, $y \rightarrow 0$
    • As $x \rightarrow \infty$, $y \rightarrow 0$
  • Critical Points: $(-1, 1)$, $(1, 1)$
  • Asymptotes: Vertical line at $x = 0$, Horizontal line at $y = 0$

Step and Constant Parent Functions

• Greatest Integer Function

  • Equation: $y = \text{int}(x) = \llbracket x \rrbracket$
  • Classification: Greatest Integer, Neither
  • Domain: $(-\infty, \infty)$
  • Range: $\{y : y \in Z\}$ (where $Z$ represents the set of integers)
  • End Behavior:
    • As $x \rightarrow -\infty$, $y \rightarrow -\infty$
    • As $x \rightarrow \infty$, $y \rightarrow \infty$
  • Critical Intervals (Steps):
    • When $x \in [-1, 0)$, $y = -1$
    • When $x \in [0, 1)$, $y = 0$
    • When $x \in [1, 2)$, $y = 1$

• Constant Function

  • Equation: $y = C$ (Example provided: $y = 2$)
  • Classification: Constant, Even
  • Domain: $(-\infty, \infty)$
  • Range: $\{y : y = C\}$
  • End Behavior:
    • As $x \rightarrow -\infty$, $y \rightarrow C$
    • As $x \rightarrow \infty$, $y \rightarrow C$
  • Critical Points: $(-1, C)$, $(0, C)$, $(1, C)$

Function Transformation Rules

• Vertical Shifts

  • $g(x) = f(x) + c$: Shifts the graph upward by $c$ units.
  • $g(x) = f(x) - c$: Shifts the graph downward by $c$ units.

• Horizontal Shifts

  • $g(x) = f(x + c)$: Shifts the graph to the left by $c$ units.
  • $g(x) = f(x - c)$: Shifts the graph to the right by $c$ units.

• Reflections

  • $g(x) = -f(x)$: Flips/reflects the graph over the $x$-axis.
  • $g(x) = f(-x)$: Flips/reflects the graph over the $y$-axis.

• Vertical Stretch or Compression

  • General form: $g(x) = c \times f(x)$
  • If $c > 1$: Stretches the function vertically.
  • If $0 < c < 1$: Compresses the function vertically.

• Horizontal Stretch or Compression

  • General form: $g(x) = f(c \times x)$
  • If $c > 1$: Compresses the function horizontally.
  • If $0 < c < 1$: Stretches the function horizontally.