Quadratic Functions Review and Analysis

Quadratic Function Analysis in Vertex Form

• Function equation: $f(x) = 2(x+5)^2 + 1$
• Geometric transformations: Vertical stretch by a factor of $2$, horizontal shift left $5$ units, and vertical shift up $1$ unit
• Vertex: $(-5, 1)$
• Axis of symmetry: $x = -5$
• Extrema: Minimum value of $1$ at $x = -5$
• Intercepts:
  • $y$-intercept: $(0, 51)$
• Domain and range:
  • Domain: $(-\bigcirc, +\bigcirc)$
  • Range: $[1, +\bigcirc)$
• Function behavior:
  • Increasing interval: $(-5, +\bigcirc)$
  • Decreasing interval: $(-\bigcirc, -5)$
  • End behavior: As $x \rightarrow -\bigcirc$, $f(x) \rightarrow +\bigcirc$; as $x \rightarrow +\bigcirc$, $f(x) \rightarrow +\bigcirc$
• Plotted coordinate points:
  • $(-7, 9)$
  • $(-6, 3)$
  • $(-5, 1)$
  • $(-4, 3)$
  • $(-3, 9)$

Quadratic Function Analysis in Intercept Form

• Function equation: $f(x) = -\frac{1}{2}(x+6)(x-2)$
• Vertex: $(-2, 8)$
• Axis of symmetry: $x = -2$
• Extrema: Maximum value of $8$ at $x = -2$
• Intercepts:
  • $x$-intercepts: $(-6, 0)$ and $(2, 0)$
  • $y$-intercept: $(0, 6)$
• Domain and range:
  • Domain: $(-\bigcirc, +\bigcirc)$
  • Range: $(-\bigcirc, 8]$
• Function behavior:
  • Increasing interval: $(-\bigcirc, -2)$
  • Decreasing interval: $(-2, +\bigcirc)$
  • End behavior: As $x \rightarrow -\bigcirc$, $f(x) \rightarrow -\bigcirc$; as $x \rightarrow +\bigcirc$, $f(x) \rightarrow -\bigcirc$
• Plotted coordinate points:
  • $(-6, 0)$
  • $(-4, 6)$
  • $(-2, 8)$
  • $(0, 6)$
  • $(2, 0)$

Additional Computational Records

• Algebraic expression: $-310-47 +3=-5$
• Numeric and character notations:
  • $1-2+(-2)$
  • 湯
  • $-0^{\circ}412/03-30$
• Isolated mathematical fragments:
  • $) (-3)$
  • $8/0=7547$
  • $(K-73-4$