Pre-Calculus: Graphs of Functions & Graphing Techniques

Pre-Calculus Mathematics MA-109: Chapter 2.6 & 2.7 Notes

Introduction

• Instructor: Prof. Mike Fitzmaurice

• Contact: mgfitzmaurice@adjuncts.pccc.edu

• Topics Covered:

  • Chapter 2.6: Graphs of Basic Functions

  • Chapter 2.7: Graphing Techniques

Continuity (Informal Definition)

• Definition: A function is continuous over an interval of its domain if its hand-drawn graph over that interval can be sketched without lifting the pencil from the paper.

  • An open

Introduction

• Instructor: Prof. Mike Fitzmaurice

• Contact: mgfitzmaurice@adjuncts.pccc.edu

• Topics Covered:

  • Chapter 2.6: Graphs of Basic Functions

  • Constant, Identity, Absolute Value, Quadratic, Cubic, Square Root, Cube Root, Reciprocal Functions

  • Chapter 2.7: Graphing Techniques

  • Vertical and Horizontal Shifts, Reflections across axes, Vertical and Horizontal Stretches/Compressions

Continuity (Informal Definition)

• Definition: A function is continuous over an interval of its domain if its hand-drawn graph over that interval can be sketched without lifting the pencil from the paper.

Chapter 2.6: Graphs of Basic Functions (Study Guide)

Review the shape, domain, range, and key features of each parent function:

• Constant Function

  • $f(x) = c$

  • Graph: Horizontal line

  • Domain: $(-\infty, \infty)$

  • Range: $[c, c]$

  • Examples: $f(x) = 3$

• Identity Function (Linear)

  • $f(x) = x$

  • Graph: Line passing through the origin with a slope of 1

  • Domain: $(-\infty, \infty)$

  • Range: $(-\infty, \infty)$

• Absolute Value Function

  • $f(x) = |x|$

  • Graph: V-shaped, symmetric about the y-axis, vertex at $(0,0)$

  • Domain: $(-\infty, \infty)$

  • Range: $[0, \infty)$

• Quadratic Function

  • $f(x) = x^2$

  • Graph: Parabola, symmetric about the y-axis, vertex at $(0,0)$

  • Domain: $(-\infty, \infty)$

  • Range: $[0, \infty)$

• Cubic Function

  • $f(x) = x^3$

  • Graph: S-shaped, symmetric about the origin, passes through $(0,0)$

  • Domain: $(-\infty, \infty)$

  • Range: $(-\infty, \infty)$

• Square Root Function

  • $f(x) = \sqrt{x}$

  • Graph: Starts at $(0,0)$ and extends to the right

  • Domain: $[0, \infty)$

  • Range: $[0, \infty)$

• Cube Root Function

  • $f(x) = \sqrt[3]{x}$

  • Graph: S-shaped, defined for all real numbers, symmetric about the origin, passes through $(0,0)$

  • Domain: $(-\infty, \infty)$

  • Range: $(-\infty, \infty)$

• Reciprocal Function

  • $f(x) = \frac{1}{x}$

  • Graph: Hyperbola with vertical asymptote at $x=0$ and horizontal asymptote at $y=0$

  • Domain: $(-\infty, 0) \cup (0, \infty)$

  • Range: $(-\infty, 0) \cup (0, \infty)$

Chapter 2.7: Graphing Techniques (Study Guide)

Understand how transformations alter the graph of a parent function $y = f(x)$.

• Vertical Shifts

  • $y = f(x) + k$

  • Shifts graph upward $k$ units if k > 0

    • Example: For $f(x) = x^2$, the graph of $g(x) = x^2 + 3$ shifts the parabola 3 units up from its original position.

  • Shifts graph downward $k$ units if k < 0

    • Example: For $f(x) = x^2$, the graph of $g(x) = x^2 - 2$ shifts the parabola 2 units down from its original position.

• Horizontal Shifts

  • $y = f(x - h)$

  • Shifts graph right $h$ units if h > 0

    • Example: For $f(x) = x^2$, the graph of $g(x) = (x - 4)^2$ shifts the parabola 4 units to the right.

  • Shifts graph left $h$ units if h < 0

    • Example: For $f(x) = x^2$, the graph of $g(x) = (x + 1)^2$ shifts the parabola 1 unit to the left.

• Reflections

  • Across x-axis: $y = -f(x)$

  • Multiplies y-coordinates by $-1$

  • Example: For $f(x) = x^2$, the graph of $g(x) = -x^2$ flips the parabola upside down, opening downwards.

  • Across y-axis: $y = f(-x)$

  • Multiplies x-coordinates by $-1$

  • Example: For $f(x) = \sqrt{x}$, the graph of $g(x) = \sqrt{-x}$ takes the original graph (which extends right from the origin) and reflects it to the left side of the y-axis.

• Vertical Stretches and Compressions

  • $y = a \cdot f(x)$

  • Vertical Stretch if |a| > 1

    • Example: For $f(x) = x^2$, the graph of $g(x) = 2x^2$ makes the parabola appear narrower, stretching it vertically by a factor of 2.

  • Vertical Compression if 0 < |a| < 1

    • Example: For $f(x) = x^2$, the graph of $g(x) = \frac{1}{2}x^2$ makes the parabola appear wider, compressing it vertically by a factor of $\frac{1}{2}$.

  • If a < 0, there is also a reflection across the x-axis.

• Horizontal Stretches and Compressions

  • $y = f(c \cdot x)$

  • Horizontal Compression if |c| > 1

    • Example: For $f(x) = x^2$, the graph of $g(x) = (2x)^2$ compresses the parabola horizontally by a factor of $\frac{1}{2}$, making it appear narrower.

  • Horizontal Stretch if 0 < |c| < 1

    • Example: For $f(x) = x^2$, the graph of $g(x) = (\frac{1}{2}x)^2$ stretches the parabola horizontally by a factor of 2, making it appear wider.

  • If c < 0, there is also a reflection across the y-axis.

Formula Column: Transformation Rules for $y = a f(x-h) + k$

• $|a|$: Vertical stretch (>1) or compression (<1)

• a < 0: Reflection across the x-axis

• $h$: Horizontal shift (right if h>0, left if h<0)

• $k$: Vertical shift (up if k>0, down if k<0)

Order of Transformations (When multiple are applied):

1. Horizontal Shifts ($-h$)

2. Stretches/Compressions and Reflections ($a$)

3. Vertical Shifts ($+k$)

Introduction

• Instructor: Prof. Mike Fitzmaurice

• Contact: mgfitzmaurice@adjuncts.pccc.edu

• Topics Covered:

  • Chapter 2.6: Graphs of Basic Functions

  • Constant, Identity, Absolute Value, Quadratic, Cubic, Square Root, Cube Root, Reciprocal Functions

  • Chapter 2.7: Graphing Techniques

  • Vertical and Horizontal Shifts, Reflections across axes, Vertical and Horizontal Stretches/Compressions

Continuity (Informal Definition)

• Definition: A function is continuous over an interval of its domain if its hand-drawn graph over that interval can be sketched without lifting the pencil from the paper.

Chapter 2.6: Graphs of Basic Functions (Study Guide)

Review the shape, domain, range, and key features of each parent function:

• Constant Function

  • $f(x) = c$

  • Graph: Horizontal line

  • Domain: $(-\infty, \infty)$

  • Range: $[c, c]$

  • Examples: $f(x) = 3$

• Identity Function (Linear)

  • $f(x) = x$

  • Graph: Line passing through the origin with a slope of 1

  • Domain: $(-\infty, \infty)$

  • Range: $(-\infty, \infty)$

• Absolute Value Function

  • $f(x) = |x|$

  • Graph: V-shaped, symmetric about the y-axis, vertex at $(0,0)$

  • Domain: $(-\infty, \infty)$

  • Range: $[0, \infty)$

• Quadratic Function

  • $f(x) = x^2$

  • Graph: Parabola, symmetric about the y-axis, vertex at $(0,0)$

  • Domain: $(-\infty, \infty)$

  • Range: $[0, \infty)$

• Cubic Function

  • $f(x) = x^3$

  • Graph: S-shaped, symmetric about the origin, passes through $(0,0)$

  • Domain: $(-\infty, \infty)$

  • Range: $(-\infty, \infty)$

• Square Root Function

  • $f(x) = \sqrt{x}$

  • Graph: Starts at $(0,0)$ and extends to the right

  • Domain: $[0, \infty)$

  • Range: $[0, \infty)$

• Cube Root Function

  • $f(x) = \sqrt[3]{x}$

  • Graph: S-shaped, defined for all real numbers, symmetric about the origin, passes through $(0,0)$

  • Domain: $(-\infty, \infty)$

  • Range: $(-\infty, \infty)$

• Reciprocal Function

  • $f(x) = \frac{1}{x}$

  • Graph: Hyperbola with vertical asymptote at $x=0$ and horizontal asymptote at $y=0$

  • Domain: $(-\infty, 0) \cup (0, \infty)$

  • Range: $(-\infty, 0) \cup (0, \infty)$

Chapter 2.7: Graphing Techniques (Study Guide)

Understand how transformations alter the graph of a parent function $y = f(x)$.

• Vertical Shifts

  • $y = f(x) + k$

  • Shifts graph upward $k$ units if k > 0

    • Example: For $f(x) = x^2$, the graph of $g(x) = x^2 + 3$ shifts the parabola 3 units up from its original position.

  • Shifts graph downward $k$ units if k < 0

    • Example: For $f(x) = x^2$, the graph of $g(x) = x^2 - 2$ shifts the parabola 2 units down from its original position.

• Horizontal Shifts

  • $y = f(x - h)$

  • Shifts graph right $h$ units if h > 0

    • Example: For $f(x) = x^2$, the graph of $g(x) = (x - 4)^2$ shifts the parabola 4 units to the right.

  • Shifts graph left $h$ units if h < 0

    • Example: For $f(x) = x^2$, the graph of $g(x) = (x + 1)^2$ shifts the parabola 1 unit to the left.

• Reflections

  • Across x-axis: $y = -f(x)$

  • Multiplies y-coordinates by $-1$

  • Example: For $f(x) = x^2$, the graph of $g(x) = -x^2$ flips the parabola upside down, opening downwards.

  • Across y-axis: $y = f(-x)$

  • Multiplies x-coordinates by $-1$

  • Example: For $f(x) = \sqrt{x}$, the graph of $g(x) = \sqrt{-x}$ takes the original graph (which extends right from the origin) and reflects it to the left side of the y-axis.

• Vertical Stretches and Compressions

  • $y = a \cdot f(x)$

  • Vertical Stretch if |a| > 1

    • Example: For $f(x) = x^2$, the graph of $g(x) = 2x^2$ makes the parabola appear narrower, stretching it vertically by a factor of 2.

  • Vertical Compression if 0 < |a| < 1

    • Example: For $f(x) = x^2$, the graph of $g(x) = \frac{1}{2}x^2$ makes the parabola appear wider, compressing it vertically by a factor of $\frac{1}{2}$.

  • If a < 0, there is also a reflection across the x-axis.

• Horizontal Stretches and Compressions

  • $y = f(c \cdot x)$

  • Horizontal Compression if |c| > 1

    • Example: For $f(x) = x^2$, the graph of $g(x) = (2x)^2$ compresses the parabola horizontally by a factor of $\frac{1}{2}$, making it appear narrower.

  • Horizontal Stretch if 0 < |c| < 1

    • Example: For $f(x) = x^2$, the graph of $g(x) = (\frac{1}{2}x)^2$ stretches the parabola horizontally by a factor of 2, making it appear wider.

  • If c < 0, there is also a reflection across the y-axis.

Formula Column: Transformation Rules for $y = a f(x-h) + k$

• $|a|$: Vertical stretch (>1) or compression (<1)

• a < 0: Reflection across the x-axis

• $h$: Horizontal shift (right if h>0, left if h<0)

• $k$: Vertical shift (up if k>0, down if k<0)

Order of Transformations (When multiple are applied):

1. Horizontal Shifts ($-h$)

2. Stretches/Compressions and Reflections ($a$)

3. Vertical Shifts ($+k$)