Notes on Graphing Lines, Functions, and Piecewise Functions

Graphing Line Segments over Intervals

• Concept: To graph a specific chunk or segment of a line rather than an infinite line, focus on the behavior at each endpoint of the given interval.

• Procedure:

  1. Plug in the $x$-values of the endpoints into the line's equation ($y = ext{equation of line}$).

  2. Determine the corresponding $y$-values to find the coordinates of the endpoints.

  3. Plot these endpoints.

  4. Draw the line segment between them.

• Endpoint Inclusion/Exclusion (Open vs. Closed Circles):

  • Parentheses ( ) or strict inequalities (<, >): Indicate that the endpoint is not included. This is represented by an open circle on the graph at that point.

  • Brackets [ ] or inclusive inequalities ($\le$, $\ge$): Indicate that the endpoint is included. This is represented by a closed circle on the graph at that point.

• Example 1: Graph $y = x + 5$ over the interval from $-5$ to $-1$ (implicitly, assumed to be $-5 \le x \le -1$ or similar based on context).

  • At $x = -5$: $y = -5 + 5 = 0$. Point: $(-5, 0)$.

  • At $x = -1$: The speaker initially mentioned positive one, but corrected to negative one for the interval.

  • You would plot these points and draw the segment connecting them.

• Example 2: Graph $y = -3x + 3$ over the interval from $0$ to $3$ (specifically, 0 < x \le 3).

  • At $x = 0$: $y = -3(0) + 3 = 3$. Point: $(0, 3)$. Since it's x < 1 for this function piece or 0 < x, this is an open circle at $(0, 3)$.

  • At $x = 1$: $y = -3(1) + 3 = 0$. Point: $(1, 0)$. (If interval was 0 < x \le 1)

  • At $x = 2$: $y = -3(2) + 3 = -3$. Point: $(2, -3)$ (if interval was 0 < x \le 2)

  • At $x = 3$: $y = -3(3) + 3 = -6$. Point: $(3, -6)$. Since it's $x \le 3$, this is a closed circle at $(3, -6)$.

  • Connect the points from $(0, 3)$ (open) to $(3, -6)$ (closed).

• Example 3: Graph $y = 2x$ over the interval from $-3$ to $1$ (implicitly, assumed to be -3 \le x < 1 or similar).

  • At $x = -3$: $y = 2(-3) = -6$. Point: $(-3, -6)$. If the interval is $\ge -3$, this is a closed circle.

  • At $x = 1$: $y = 2(1) = 2$. Point: $(1, 2)$. If the interval is < 1, this is an open circle.

  • Plot additional points within the interval for clarity, e.g., $(0, 0)$, $(-1, -2)$.

Administrative Announcements

• Homework Due Dates:

  • Online homework for Section $3.3$: Due tonight (was mistakenly set to September $29$, corrected to today).

  • Written homework for Section $3.3$: Due next Tuesday (September $26$).

• Optional Exam One Post-Wrapper Assignment:

  • Purpose: A reflective assignment to help students analyze their performance on Exam One. It's especially useful as the final exam is cumulative.

  • Task: Review each exam question, identify its topic (e.g., "finding the slope of a perpendicular line"), note points earned, assess skill mastery (solid skill / need work), and add brief notes (e.g., "didn't have the formula," "need help with blank").

  • Format: PDF and Word copies available for download.

  • Value: Worth $10$ points in the "test category" (a new category). It won't significantly boost a grade but helps with learning and identifying areas for improvement.

  • Due Date: September $27$.

  • Recommendation: If you did well, it's optional; if you need to improve, it's highly recommended for growth and future success.

  • Importance: Math builds on previous concepts. Understanding missed topics now is crucial for later material and the cumulative final exam.

The Library of Functions

• Objective: To memorize eight common basic graphs in their most original form. These form the foundation for understanding transformations (shifting, flipping, stretching, shrinking) on Wednesday.

• Memorization Strategy: Practice leads to memorization. A note card will be allowed on the next exam for these basic graphs.

• Required Points for Plotting: At least three points should be plottable for each function.

1. Constant Function

   • Equation: $y = b$

   • Description: A horizontal line.

   • Key Points: Any three points with the same $y$-coordinate, e.g., $(-1, b)$, $(0, b)$, $(1, b)$.

   • Domain: $(-\infty, \infty)$

   • Range: $[b]$ (single value)

   • Characteristic: Output (y-value) is constant regardless of input (x-value).

2. Identity Function

   • Equation: $y = x$

   • Description: A line with a slope of $1$ and a y-intercept of $0$.

   • Key Points: $(0, 0)$, $(1, 1)$, $(-1, -1)$.

   • Domain: $(-\infty, \infty)$

   • Range: $(-\infty, \infty)$

   • Characteristic: Input equals output. No restrictions on domain or range.

3. Square Function (Parabola / Quadratic)

   • Equation: $y = x^2$

   • Description: A U-shaped curve, symmetric about the y-axis.

   • Key Points: $(0, 0)$, $(1, 1)$, $(-1, 1)$. (Can also plot $(2, 4)$, $(-2, 4)$).

   • Domain: $(-\infty, \infty)$

   • Range: $[0, \infty)$

   • Characteristic: The lowest point (vertex) is at $(0, 0)$, and all y-values are non-negative.

4. Cube Function

   • Equation: $y = x^3$

   • Description: An S-shaped curve, passing through the origin, symmetric about the origin.

   • Key Points: $(0, 0)$, $(1, 1)$, $(-1, -1)$. (Can also plot $(2, 8)$, $(-2, -8)$).

   • Domain: $(-\infty, \infty)$

   • Range: $(-\infty, \infty)$

   • Characteristic: Can take any real number as input and produce any real number as output.

5. Square Root Function

   • Equation: $y = \sqrt{x}$

   • Description: Starts at the origin and extends to the right, forming a curve.

   • Key Points: $(0, 0)$, $(1, 1)$, $(4, 2)$.

   • Domain: $[0, \infty)$

   • Range: $[0, \infty)$

   • Characteristic: Cannot take the square root of negative numbers (real numbers context), hence the restricted domain. Outputs are always non-negative.

6. Cube Root Function

   • Equation: $y = \sqrt[3]{x}$

   • Description: A horizontal S-shaped curve passing through the origin, symmetric about the origin.

   • Key Points: $(0, 0)$, $(1, 1)$, $(-1, -1)$. (Can also plot $(8, 2)$, $(-8, -2)$).

   • Domain: $(-\infty, \infty)$

   • Range: $(-\infty, \infty)$

   • Characteristic: Can take cube roots of negative numbers, zero, or positive numbers. No restrictions on domain or range.

7. Reciprocal Function

   • Equation: $y = \frac{1}{x}$

   • Description: Two separate curves in quadrants I and III, approaching but never touching the x and y axes (asymptotes).

   • Key Points: $(1, 1)$, $(\frac{1}{2}, 2)$, $(2, \frac{1}{2})$, $(-1, -1)$, $(-\frac{1}{2}, -2)$, $(-2, -\frac{1}{2})$.

   • Domain: $(-\infty, 0) \cup (0, \infty)$ (all real numbers except $0$).

   • Range: $(-\infty, 0) \cup (0, \infty)$ (all real numbers except $0$).

   • Characteristic: Division by zero is undefined, so $x \ne 0$. Consequently, the output $y$ can never be zero.

8. Absolute Value Function

   • Equation: $y = |x|$

   • Description: A V-shaped graph with its vertex at the origin, opening upwards, symmetric about the y-axis.

   • Key Points: $(0, 0)$, $(1, 1)$, $(-1, 1)$.

   • Domain: $(-\infty, \infty)$

   • Range: $[0, \infty)$

   • Characteristic: The absolute value operation makes any non-zero input positive, resulting in all y-values being non-negative. The entire graph is above or on the x-axis.

Piecewise Defined Functions

• Definition: A function defined by different equations (or "pieces") over different parts of its domain.

• Structure: It's still a single function, meaning for every input ($x$), there is only one output ($y$). This is ensured by careful definition of the intervals and endpoint inclusion.

• How to Read/Decode: Each line specifies an equation and the interval over which that equation applies.

• Graphing Procedure:

  1. Graph each equation as if it were a full line/curve.

  2. Keep only the portion of the graph that falls within its specified interval.

  3. Pay close attention to the endpoints of each interval, using open circles for non-inclusive limits (<, > or parentheses) and closed circles for inclusive limits ($\le, \ge$ or brackets).

  4. Combine all the segments/curves onto a single coordinate plane to form the complete graph of the piecewise function.

• Example of Reading and Graphing: $f(x) = \begin{cases} 3 &amp; \text{if } x \le 0 \ -x &amp; \text{if } x &gt; 0 \end{cases}$

  • Piece 1: $y = 3$ for $x \le 0$

    • This is a horizontal line at $y=3$. It applies for all $x$ from $(-\infty, 0]$ (inclusive of $0$).

    • Endpoint at $x=0$: $y=3$. Plot $(0, 3)$ as a closed circle.

    • The line extends left from $(0, 3)$.

  • Piece 2: $y = -x$ for x > 0

    • This is a line with slope $-1$ and y-intercept $0$. It applies for all $x$ from $(0, \infty)$.

    • Endpoint at $x=0$: $y=0$. Plot $(0, 0)$ as an open circle (since x > 0).

    • At $x=1$: $y=-1$. Plot $(1, -1)$.

    • The line extends right from $(0, 0)$ (open circle).

• Analyzing a Piecewise Defined Function (Example): $f(x) = \begin{cases} \frac{2}{3}x + 1 &amp; \text{if } x &lt; 1 \ -2x + 1 &amp; \text{if } x \ge 1 \end{cases}$

  • Finding Function Values:

    • $f(0)$: Since 0 < 1, use the top rule: $f(0) = \frac{2}{3}(0) + 1 = 1$. So, $f(0) = 1$.

    • $f(1)$: Since $1 \ge 1$, use the bottom rule: $f(1) = -2(1) + 1 = -1$. So, $f(1) = -1$.

    • $f(2)$: Since $2 \ge 1$, use the bottom rule: $f(2) = -2(2) + 1 = -4 + 1 = -3$. So, $f(2) = -3$.

  • Finding the Domain:

    • The first interval is $(-\infty, 1)$.

    • The second interval is $[1, \infty)$.

    • When combined, $(-\infty, 1) \cup [1, \infty)$ covers all real numbers.

    • Domain: $(-\infty, \infty)$.

  • Finding Intercepts:

    • Y-intercept: This occurs when $x = 0$. Since 0 < 1, we use the top rule: $y = \frac{2}{3}(0) + 1 = 1$. The y-intercept is $(0, 1)$.

    • X-intercepts: This occurs when $y = 0$. You must check each piece. Looking at the graph, the x-intercept is within the first piece (x < 1).

      • For the first piece ($y = \frac{2}{3}x + 1$): Set $0 = \frac{2}{3}x + 1 \implies -1 = \frac{2}{3}x \implies x = -\frac{3}{2} = -1.5$. Since -1.5 < 1, this is a valid x-intercept: $(-1.5, 0)$.

      • For the second piece ($y = -2x + 1$): Set $0 = -2x + 1 \implies 2x = 1 \implies x = \frac{1}{2}$. However, this piece is defined for $x \ge 1$. Since $\frac{1}{2}$ is not $\ge 1$, this point is NOT an x-intercept for this function.

  • Finding the Range:

    • Graphing helps visualize the range.

    • Piece 1: $y = \frac{2}{3}x + 1$ for x < 1

      • At $x=1$ (open circle): $y = \frac{2}{3}(1) + 1 = \frac{5}{3}$. The line goes downwards to the left from $(1, 5/3)$ (open circle).

      • Range for this piece: $(-\infty, \frac{5}{3})$.

    • Piece 2: $y = -2x + 1$ for $x \ge 1$

      • At $x=1$ (closed circle): $y = -2(1) + 1 = -1$. The line goes downwards to the right from $(1, -1)$ (closed circle).

      • Range for this piece: $(-\infty, -1]$.

    • Combined Range: The function covers all y-values from negative infinity up to $\frac{5}{3}$. The 'gap' between $-1$ and $\frac{5}{3}$ for $y$ is covered by the first piece. So the highest y-value is $\frac{5}{3}$ (not included).

    • Range: $(-\infty, \frac{5}{3})$.