Trig and geometry Ultimate Study Guide

General Transformation Model for Sinusoidal Functions

• The standard mathematical models for transformed sinusoidal functions are defined as:   $y = a \sin(b(x - c)) + d$   and   $y = a \cos(b(x - c)) + d$

• The parameters $a$, $b$, $c$, and $d$ each control a distinct geometric transformation of the parent curve:

  • Vertical Stretch and Shrink ($a$):

  • When $a > 1$, the graph undergoes a vertical stretch by a factor of $a$.

  • When $0 < a < 1$, the graph undergoes a vertical shrink (compression) by a factor of $a$.

  • The amplitude of the function is defined by the absolute value: $\text{Amplitude} = |a|$.

  • If the parameter is preceded by a negative sign ($-a$), the graph is reflected across the $x\text{-axis}$.

  • Horizontal Stretch and Shrink ($b$):

  • When $b > 1$, the graph undergoes a horizontal shrink by a factor of $\frac{1}{b}$.

  • When $0 < b < 1$, the graph undergoes a horizontal stretch by a factor of $\frac{1}{b}$.

  • The period of the sinusoidal function is determined using the parameter $b$ through the relation:       $\text{Period} = \frac{2\pi}{b}$

  • Horizontal Translation or Phase Shift ($c$):

  • In the standard factored expression $b(x - c)$, the value $c$ represents the phase shift.

  • When $c > 0$ (appearing as $(x - c)$), the phase shift moves horizontally to the right by $c$ units.

  • When $c < 0$ (appearing as $(x + c)$), the phase shift moves horizontally to the left by $|c|$ units.

  • In the expanded form $y = a \sin(bx - bc)$ or $y = a \cos(bx - bc)$, the horizontal displacement is found by dividing the constant term by the frequency coefficient: $\text{Phase Shift} = \frac{bc}{b} = c$.

  • Vertical Translation ($d$):

  • When $d > 0$, the graph is vertically shifted upward by $d$ units.

  • When $d < 0$, the graph is vertically shifted downward by $|d|$ units.

  • The vertical translation determines the midline of the oscillating function, which is the horizontal line:       $\text{Midline}: y = d$

Describing Transformations of Trigonometric Functions

• Transformation Analysis for $y = 5 \sin(x - 1) + 2$:

  • Vertical stretch: The amplitude factor $a = 5$ ($5 > 1$) stretches the graph vertically by a factor of $5$.

  • Horizontal translation: The term $(x - 1)$ gives $c = 1$, shifting the graph to the right by $1$ unit.

  • Vertical translation: The term $+ 2$ gives $d = 2$, shifting the graph upward by $2$ units to a new midline at $y = 2$.

• Transformation Analysis for $y = -\cos(2(x + \pi)) - 1$:

  • Reflection: The leading negative sign reflects the graph across the $x\text{-axis}$.

  • Horizontal shrink: The parameter $b = 2$ compresses the graph horizontally by a factor of $\frac{1}{2}$, resulting in a period of:     $\text{Period} = \frac{2\pi}{2} = \pi$

  • Horizontal translation: The term $(x + \pi)$ corresponds to $c = -\pi$, resulting in a phase shift to the left by $\pi$ units.

  • Vertical translation: The parameter $d = -1$ shifts the graph downward by $1$ unit to a midline at $y = -1$.

• Transformation Analysis for $y = \cos(x - 3) + 2$:

  • Horizontal translation: The term $(x - 3)$ shifts the curve to the right by $3$ units.

  • Vertical translation: The constant term $+ 2$ shifts the curve upward by $2$ units, establishing the midline at $y = 2$.

• Transformation Analysis for $y = -3 \sin(x) + 4$:

  • Reflection: The negative coefficient causes a vertical reflection across the $x\text{-axis}$.

  • Vertical stretch: The coefficient magnitude yields an amplitude of $|-3| = 3$, creating a vertical stretch by a factor of $3$.

  • Vertical translation: The constant $+ 4$ shifts the function upward by $4$ units to a midline at $y = 4$.

Determining Amplitude, Period, and Phase Shift

• Function: $y = 3 \sin\left(\frac{1}{2}(x - \pi)\right)$

  • Amplitude:     $\text{Amplitude} = |a| = |3| = 3$

  • Period:     $\text{Period} = \frac{2\pi}{b} = \frac{2\pi}{\frac{1}{2}} = 2\pi \times 2 = 4\pi$

  • Phase Shift:     $\text{Phase Shift} = \pi \text{ units to the right}$

• Function: $y = 4 \cos(\pi(x + 2))$

  • Amplitude:     $\text{Amplitude} = |a| = |4| = 4$

  • Period:     $\text{Period} = \frac{2\pi}{b} = \frac{2\pi}{\pi} = 2$

  • Phase Shift:     $\text{Phase Shift} = 2 \text{ units to the left}$

• Function: $y = -3 \cos\left(2\left(x + \frac{\pi}{2}\right)\right)$

  • Amplitude:     $\text{Amplitude} = |a| = |-3| = 3$

  • Period:     $\text{Period} = \frac{2\pi}{b} = \frac{2\pi}{2} = \pi$

  • Phase Shift:     $\text{Phase Shift} = \frac{\pi}{2} \text{ units to the left}$

• Function: $y = -2 \sin\left(\frac{1}{2}(x + \pi)\right)$

  • Amplitude:     $\text{Amplitude} = |a| = |-2| = 2$

  • Period:     $\text{Period} = \frac{2\pi}{b} = \frac{2\pi}{\frac{1}{2}} = 4\pi$

  • Phase Shift:     $\text{Phase Shift} = \pi \text{ units to the left (or } \frac{\pi}{2} \text{ units depending on factoring)}$

Graphing Sinusoidal Functions

• Analysis and Graphing Parameters for $f(x) = 4 \sin\left(3\left(x + \frac{\pi}{6}\right)\right) - 1$:

  • Amplitude: $|a| = |4| = 4$

  • Midline: $y = d = -1$

  • Period:     $\text{Period} = \frac{2\pi}{b} = \frac{2\pi}{3}$

  • Phase Shift:     $\text{Phase Shift} = \frac{\pi}{6} \text{ units to the left}$

  • Range and Key Values:

  • Maximum value: $y = -1 + 4 = 3$

  • Minimum value: $y = -1 - 4 = -5$

  • Horizontal grid markers and critical phase boundaries along the domain include:       $-\frac{11\pi}{6}$       $-\frac{5\pi}{6}$       $-\frac{3\pi}{6}$       $\frac{5\pi}{6}$       $\frac{7\pi}{6}$

• Analysis and Graphing Parameters for $f(x) = -5 \cos\left(2\left(x - \frac{\pi}{3}\right)\right) + 3$:

  • Amplitude: $|a| = |-5| = 5$

  • Midline: $y = d = 3$

  • Period:     $\text{Period} = \frac{2\pi}{b} = \frac{2\pi}{2} = \pi$

  • Phase Shift:     $\text{Phase Shift} = \frac{\pi}{3} \text{ units to the right}$

  • Range and Key Values:

  • Due to the negative leading coefficient ($a = -5$), the cosine curve reflects across the midline.

  • Maximum value: $y = 3 + 5 = 8$

  • Minimum value: $y = 3 - 5 = -2$

  • Significant axis markings and coordinate points include:       $-\frac{23\pi}{6}$       $\frac{11\pi}{2}$       $3\pi$       $-3$       $3$

Writing Trigonometric Equations from Given Characteristics

• Constructing Cosine Equation 1:

  • Given characteristics:

  • Base function: Cosine

  • Amplitude: $3$

  • Period: $\pi$

  • Phase shift: $\frac{3\pi}{2}$

  • Vertical shift: $2$ units up

  • Calculation of frequency parameter $b$:     $\text{Period} = \frac{2\pi}{b}$     $\pi = \frac{2\pi}{b}$     $\pi b = 2\pi$     $b = 2$

  • Assembly of parameter values:     $a = 3$     $b = 2$     $c = \frac{3\pi}{2}$     $d = 2$

  • Resulting equation:     $y = 3 \cos\left(2\left(x - \frac{3\pi}{2}\right)\right) + 2$

• Constructing Sine Equation 2:

  • Given characteristics:

  • Base function: Sine

  • Amplitude: $2$

  • Period: $6$

  • Phase shift: $-\frac{\pi}{4}$

  • Calculation of frequency parameter $b$:     $\text{Period} = \frac{2\pi}{b}$     $6 = \frac{2\pi}{b}$     $6b = 2\pi$     $b = \frac{2\pi}{6} = \frac{\pi}{3}$

  • Assembly of parameter values:     $a = 2$     $b = \frac{\pi}{3}$     $c = -\frac{\pi}{4}$     $d = 0$

  • Resulting equation:     $y = 2 \sin\left(\frac{\pi}{3}\left(x + \frac{\pi}{4}\right)\right)$

• Constructing Sine Equation 3:

  • Given characteristics:

  • Base function: Sine

  • Amplitude: $1$

  • Period: $2\pi$

  • Phase shift: $\frac{\pi}{3}$

  • Midline: $-3$

  • Reflection: Reflected over the $x\text{-axis}$

  • Calculation of frequency parameter $b$:     $\text{Period} = \frac{2\pi}{b}$     $2\pi = \frac{2\pi}{b}$     $b = 1$

  • Assembly of parameter values:     $a = -1$     $b = 1$     $c = \frac{\pi}{3}$     $d = -3$

  • Resulting equation:     $y = -\sin\left(x - \frac{\pi}{3}\right) - 3$

• Constructing Cosine Equation 4:

  • Given characteristics:

  • Base function: Cosine

  • Amplitude: $5$

  • Period: $\frac{3\pi}{4}$

  • Phase shift: $\frac{\pi}{3}$

  • Midline: $4$

  • Calculation of frequency parameter $b$:     $\text{Period} = \frac{2\pi}{b}$     $\frac{3\pi}{4} = \frac{2\pi}{b}$     $3\pi b = 8\pi$     $b = \frac{8\pi}{3\pi} = \frac{8}{3}$

  • Assembly of parameter values:     $a = 5$     $b = \frac{8}{3}$     $c = \frac{\pi}{3}$     $d = 4$

  • Resulting equation:     $y = 5 \cos\left(\frac{8}{3}\left(x - \frac{\pi}{3}\right)\right) + 4$