Integrated Math 3 Semester 1 Review

Integrated Math 3 Semester 1 Review Packet

Domain and Range

• Define domain and range using interval notation.
• Example intervals:
  • Domain: $(-\infty, \infty)$
  • Range: $[-5, \infty)$

Intervals

• Positive and negative intervals:
  • Positive: $(-\infty, -2) \cup (0, 1) \cup (3, \infty)$
  • Negative: $(-2, 0) \cup (1, 3)$

Increasing and Decreasing Intervals

• Increasing: $(-1.25, 1.5) \cup (2.25, \infty)$
• Decreasing: $(-\infty, -1.25) \cup (1.5, 2.25)$

X-Intercepts

• X-intercepts: $(-2, 0), (0, 0), (1, 0), (3, 0)$

End Behavior

• As $x \to -\infty$, $y \to 0$; As $x \to \infty$, $y \to 0$.

Transformations from Parent Functions

• General Form: $y = a(x - h)^3 + k$
• Examples:
  • Quadratic Transformation: $f(x) = 3(x - 4)^2 - 5$ results in upward shift and stretch.
  • Absolute Value: $g(x) = 0.5(x - 3) + 4$ reflects vertical compression.

Writing Equations

• Examples:
  • Quadratic with reflection: $y = -1(x + 7)(x)$
  • Absolute value with horizontal shift and compression: $y = 0.5|x - 3|$
  • Quartic with vertical stretch: $y = -8x^4$.

Functions and Their Transformations

• Given $y = x$:
  • $f(x) = 3(x - 2) + 4$ (vertical stretch by 3, shift right 2, up 4)
  • $g(x) = (x + 1) + 2$ (shift left, up 2)
  • $h(x) = -2(x - 7)$ (reflect over x-axis, stretch by a factor of 2, shift right 7)

End Behavior Table

• Notation for polynomial end behavior.
• Even degrees with positive coefficients rise to infinity on both sides.

Leading Coefficient Test

• Determines end behavior based on degree and leading coefficient.
• Example:
  • For $g(x) = -3x + 5x^3 - 2x - 7$, as $x \to \infty$, $y \to -\infty$.

Transformations Impacting Functions

• Do transformations affect end behavior, domain, range? Explain with examples:
  • For a vertical stretch: $g(x) = 3x^4$ (no change in domain or range, only affects steepness of the curve).

Sketching Functions

• Sketch the parent function and the transformed function to illustrate characteristics.