Test 3 Notes

Review of Functions and Graphs

Question 1: Relation as a Function

• Definition of a Function: A relation is a function if each input (x-value) has exactly one output (y-value).

• Vertical Line Test: A visual test to determine if a relation is a function; if a vertical line intersects the graph at more than one point, it is not a function.

• Example of Relation Analysis: For given coordinates (e.g. (2, 13), (1, 1), (10, 21)), check if each input has a unique output.

Domain and Range

• Domain: Set of all possible input values (x).

• Range: Set of all possible output values (y).

Question 2: Comparison of Functions

• Given Function: $f(x) = 2x + 6$

• To understand how $g(x)$ compares to $f(x)$, analyze transformations:

  • a) $g(x) = 2(x + 8) + 6$

  • Transformation: Horizontal shift left by 8 units.

  • Slope: Same as $f(x)$, which is 2.

  • Y-intercept: Calculate:
    $g(0) = 2(0 + 8) + 6 = 16$.

  • b) $g(x) = (2x + 6) - 7$

  • Transformation: Vertical shift down by 7 units.

  • Slope: 2.

  • Y-intercept: Calculate:
    $g(0) = 2(0) + 6 - 7 = -1$.

  • c) $g(x) = 2(x - 5) + 6$

  • Transformation: Horizontal shift right by 5 units.

  • Slope: 2.

  • Y-intercept: Calculate:
    $g(0) = 2(0 - 5) + 6 = -4$.

  • d) $g(x) = 4.5(2x + 6)$

  • Transformation: Vertical stretch by a factor of 4.5.

  • Slope: $2 imes 4.5 = 9$.

  • Y-intercept: Calculate:
    $g(0) = 4.5(0 + 6) = 27$.

  • e) $g(x) = 2(3.5x) + 6$

  • Transformation: Slope changed to $7$; no horizontal or vertical shifts.

  • Y-intercept: Calculate:
    $g(0) = 6$.

  • f) $g(x) = 0.5(2x + 6)$

  • Transformation: Vertical compression by a factor of 0.5.

  • Slope: $1$.

  • Y-intercept: Calculate:
    $g(0) = 0.5(0 + 6) = 3$.

Question 3: Transformation Values of k and h

• a) Given: $g(x) = f(x) + k$. Choose any value for $k$ (e.g. $k = c$, where $c$ is a constant).

• b) Given: $g(x) = f(x - h)$. Choose any value for $h$ (e.g. $h = d$, where $d$ is a constant).

Question 4: New Function Expressions based on f(x)

• Given $f(x) = -x + 1$, create new functions:

  • a) $g(x) = f(x - 3)$

  • b) $h(x) = -5f(x)$

  • c) $p(x) = f(4x)$

  • d) $m(x) = f(x + 2)$

  • e) $n(x) = f(x) - 8$

  • f) $q(x) = 3f(x)$

  • g) $t(x) = f(x) + 6$

  • h) $k(x) = f(x - 14)$

Question 5: Evaluating the Function f(x)

• Given: $f(x) = 4x$

• Find the following:

  • a) $f(-3) = 4 imes (-3) = -12$

  • b) $f(- rac{1}{4}) = 4 imes (- rac{1}{4}) = -1$

Question 6: Maria's Bracelet Production

Data Table

• a) Linear Equation: The form of the equation can be modeled as $y = mx + b$, using the given coordinates to find $m$ (slope) and $b$ (y-intercept).

• b) Domain: Reasonable domain values (e.g., $x ext{ is in } [0, T]$ where $T$ is the maximum hours Maria can work).

• c) Calculate bracelets at 5.5 hours:

  • If function is linear, use the slope to estimate the number of bracelets made at the given time.

Question 7: Finding Linear Equation Rule

• Develop both recursive and explicit formulas from given data.

• Explicit Formula: For example, $y = mx + b$ where m is slope and b y-intercept.

• Recursive Formula: Establish methods to determine specific sequence terms based on previous terms.

Question 8: Analyzing Arithmetic Sequences

• First sequence formula: $a_n = -3n + 9$

  • First Term: Evaluate when $n = 1$ → $a_1 = -3(1) + 9 = 6$.

  • Common Difference: Calculate n iterations to ascertain the difference as $6$.

  • Recursive Equation: $a_n = a_{n-1} + d$ where d is common difference.

• Second sequence formula: $a_n = a_{n-1} + 2, a_1 = -4$

  • Common Difference: $2$.

  • Explicit Equation: Show form based on general n terms.

• Third sequence formula: $a_n = 15 + 4n$

  • First Term: When $n = 1$ → $19$

  • Common Difference: $4$.

  • Recursive Equation:

  • Find components similar to above.

Question 12: Jamal's Shelf Production

• Analyze working hours and production data using possible linearity in the function.

• Function S(h): $S(h) = mh + b$

Question 13: Analyzing Ice Cream Sales

• Given Data: Relation between temperature and ice cream cones sold.

Table of Data

• a) Scatter Plot: Draw and label axes.

• b) Equation of the line: Determine slope, (change in ice cream cones/change in temperature).

• c) Interpret the slope: Reflects change in cones sold per degree temperature.

• d) Correlation Type:

  • Positive Correlation: As temperature increases, ice cream sales increase as well.

Question 14: Analyzing Association in Graphs

• Based on correlation coefficient $r = 0.9509$, interpret strength of the relationship.

• Estimation of Ice Cream Sales:

  • For 90°F: Extrapolation (projecting beyond known values).

  • For 70°F: Interpolation (within known data points).

• Causation Clarification: Can correlation suggest a causal relationship?