Algebra Fundamentals and Function Transformations Study Guide

Intervals for Positive and Negative Function Values

• The sign of a linear function $g(x) = 5x - 12$ is determined by solving for when the output is greater than or less than zero.
• Positive Intervals: To find where the function is positive, solve the inequality $5x - 12 > 0$. Adding 12 to both sides gives $5x > 12$, and dividing by 5 results in $x > \frac{12}{5}$. In interval notation, the function is positive on $(\frac{12}{5}, \infty)$.
• Negative Intervals: To find where the function is negative, solve the inequality $5x - 12 < 0$. This results in $5x < 12$, or $x < \frac{12}{5}$. In interval notation, the function is negative on $(-\infty, \frac{12}{5})$.

Piecewise-Defined Functions in Real-World Scenarios

• Piecewise functions are used to model situations where different rules apply to different intervals of the domain.
• Scenario: Gordon's Summer Job Earnings
  • Gordon earns a base rate of $\$10/h$ for the first $40$ hours worked in a week.
  • For any leisure or overtime hours exceeding $40$, he receives $1.5$ times his hourly pay, which equals $\$15/h$.
  • Function Rule for Earnings (E) based on hours worked ($n$):
    • For $0 \le n \le 40$, the earnings are defined by the equation $E(n) = 10n$.
    • For $n > 40$, the earnings include the base pay for 40 hours $(10 \times 40 = 400)$ plus the overtime rate for hours exceeding 40. This is expressed as $E(n) = 400 + 15(n - 40)$.
    • Simplified, the overtime rule is $E(n) = 15n - 200$.
  • Consolidated Function:
    • $E(n) = 10n \text{ for } 0 \le n \le 40$
    • $E(n) = 15n - 200 \text{ for } n > 40$

Geometric Transformations of Functions

Transformations modify the parent graph based on specific mathematical operations applied to the function rule.

Translations (Shifts)

• Horizontal Shifts: A shift to the left by $h$ units is represented by $f(x + h)$. A shift to the right by $h$ units is represented by $f(x - h)$.
  • If $f(x) = x^2$ is translated left $6$ units, the intermediate function is $(x + 6)^2$.
  • If a function is shifted left $4$ units, the transformation is $f(x + 4)$.
• Vertical Shifts: A shift up by $k$ units is represented by $f(x) + k$, while a shift down is represented by $f(x) - k$.
  • If $f(x) = x^2$ is translated down $5$ units, the final rule is $g(x) = (x + 6)^2 - 5$.
  • If $f(x) = x^2$ is translated left $5$ units and down $9$ units, the rule is $g(x) = (x + 5)^2 - 9$.

Reflecting Functions

• X-axis Reflection: To reflect a function $f(x)$ across the x-axis, the entire function is multiplied by $-1$, resulting in $g(x) = -f(x)$.
  • For $f(x) = x^2 + 3$, a reflection in the x-axis results in $g(x) = -(x^2 + 3) = -x^2 - 3$.
  • For $f(x) = 3x + 4$, a reflection in the x-axis results in $g(x) = -(3x + 4) = -3x - 4$.

Stretches and Compressions

• Vertical Stretch: Multiplying a function by a factor $a$ where $|a| > 1$ stretches the graph vertically.
  • If $f(x) = -6x$ is vertically stretched by a factor of $7$ and reflected in the x-axis, the stretch makes it $-42x$, and the reflection (multiplication by $-1$) results in $g(x) = 42x$.
  • If $f(x) = 3x^2$ is stretched to $g(x) = 9x^2 - 4$, the vertical stretch factor is $3$ because $3 \times (3x^2) = 9x^2$. The graph is also translated down $4$ units.
• Vertical Compression: Multiplying a function by a factor $a$ where $0 < |a| < 1$ compresses the graph vertically toward the x-axis.
  • A vertical compression of $y = x$ by a factor of $\frac{1}{4}$ results in the equation $y = \frac{1}{4}x$.

Arithmetic Operations with Rational Numbers

• Addition of Fractions: To add fractions like $\frac{7}{8} + \frac{1}{6}$, find a common denominator (24).
  • $\frac{21}{24} + \frac{4}{24} = \frac{25}{24}$.
• Addition with Mixed Numbers: For $1 \frac{8}{9} + \frac{1}{3}$, convert the mixed number to an improper fraction: $\frac{17}{9} + \frac{3}{9} = \frac{20}{9}$. Note: The reference indicates the solution for this specific format as $\frac{22}{9}$.
• Subtraction of Mixed Numbers: For $8 \frac{1}{3} - 6 \frac{1}{4}$, convert to improper fractions: $\frac{25}{3} - \frac{25}{4}$.
  • Find the common denominator (12): $\frac{100}{12} - \frac{75}{12} = \frac{25}{12} = 2 \frac{1}{12}$.
• Multiplication of Mixed Numbers: For $\frac{1}{6} \times 6 \frac{1}{4}$, convert the mixed number to an improper fraction: $\frac{1}{6} \times \frac{25}{4} = \frac{25}{24} = 1 \frac{1}{24}$.

Analyzing the Average Rate of Change

• The average rate of change for a function $f(x)$ over an interval $[a, b]$ is calculated using the formula:   $\frac{f(b) - f(a)}{b - a}$
• Calculation for $f(x) = 4 - |x|$:
  • Interval $[-2, 0]$:       $\frac{f(0) - f(-2)}{0 - (-2)} = \frac{4 - 2}{2} = 1$
  • Interval $[0, 4]$:       $\frac{f(4) - f(0)}{4 - 0} = \frac{0 - 4}{4} = -1$
  • Interval $[-2, 4]$:       $\frac{f(4) - f(-2)}{4 - (-2)} = \frac{0 - 2}{6} = -\frac{1}{3}$
• Conclusion: Because the average rates of change are not constant across different intervals, the function is nonlinear.

Characteristics of Quadratic and Absolute Value Functions

• Intercepts of Quadratics: For $f(x) = x^2 - 49$:
  • y-intercept: Set $x = 0$, so $f(0) = -49$.
  • x-intercepts: Set $f(x) = 0$, so $x^2 = 49$, which gives $x = \pm 7$.
• Increase and Decrease in Absolute Value Functions: For the function $y = -7 - |x|$:
  • The vertex is at $(0, -7)$. Because of the negative sign before the absolute value, the graph opens downward.
  • Increasing Interval: The function increases from left to right on the interval $(-\infty, 0)$.
  • Decreasing Interval: The function decreases from left to right on the interval $(0, \infty)$.
• Domain and Range: For the function $y = |x + 3| - 2$:
  • Domain: There are no restrictions on $x$, so the domain is all real numbers.
    • Set-builder: $\{x | x \text{ is a real number} \}$
    • Interval: $(-\infty, \infty)$
  • Range: The smallest value of the absolute value is $0$, which occurs at $x = -3$. Thus, the minimum value of the function is $y = -2$.
    • Set-builder: $\{y | y \ge -2 \}$
    • Interval: $[-2, \infty)$

Applied Transformations and Mapping

• Function Mapping: If $g(x) = 6f(x)$, every output value of $f(x)$ is multiplied by 6.
  • $( -7, 8 ) \rightarrow ( -7, 48 )$
  • $( -3, 3 ) \rightarrow ( -3, 18 )$
  • $( 0, -1 ) \rightarrow ( 0, -6 )$
  • $( 2, 7 ) \rightarrow ( 2, 42 )$
  • $( 10, 5 ) \rightarrow ( 10, 30 )$
• Relative Differences: The functions $y = x$ and $y = x - 5$ are related such that every output for $y = x - 5$ is exactly 5 less than the corresponding output for $y = x$. This results in a vertical translation the graph of $y = x$ down 5 units.
• Temporal Shifts: If an airplane's altitude function $f(x)$ is shifted by leaving two hours late, the transformation is horizontal. Leaving later means a delay, modeled as $y = f(x - 2)$.

Ratios, Proportions, and Percentages

• Ratios in Simplest Form: Marie saved $\$35$ and spent $\$7$.
  • Current balance: $35 - 7 = 28$.
  • Ratio of current savings to previous balance: $28 : 35$.
  • Simplified by dividing by the common factor of $7$: $4 : 5$.
• Converting Decimals to Percents: To convert $0.036$ to a percent, multiply by 100, resulting in $3.6\%$.
• Solving Percent Problems with Equations:
  • To find what percent $P$ of $380$ is $190$:       $P \times 380 = 190 \rightarrow P = \frac{190}{380} = 0.5 = 50\%$
  • In Pierce City, if $1,282$ people (those under age 20) represent $19\%$ of the total population ($T$):       $0.19 \times T = 1282 \rightarrow T = \frac{1282}{0.19} \approx 6747 \text{ people}$.
• Sales Tax: On a $\$56.70$ purchase with a $4\%$ tax rate:
  • Tax amount = $0.04 \times 56.70 = \$2.268$, rounded to $\$2.27$.
• Proportions: If a fruit stand charges $\$3$ for $4$ pounds of fruit, the cost ($C$) for $20$ pounds can be solved by setting up a proportion:
  • $\frac{3}{4} = \frac{C}{20}$
  • $4C = 60 \rightarrow C = 15$. The cost is $\$15$.