Study Notes on Basic Functions and Graphs

Introduction to Basic Functions

• Overview of the session's content, focusing on summarizing graphs of basic functions.

Linear Functions

• Equation of a Line: The slope-intercept form of a linear function is given by:

  • $y = mx + b$

  • In function notation: $f(x) = mx + b$

  • Where m represents the slope and b represents the y-intercept.

  • Slope is not zero: The graph is a slanted line.

  • Slope is zero: The equation simplifies to $f(x) = b$, equivalent to $y = b$, representing a horizontal line.

  • This type is called a constant function because the function value remains constant at b, regardless of the value of x.

Graph Analysis of Basic Functions

• Linear Function Example: $f(x) = x$

  • Characteristics:

  • Slope of 1.

  • y-intercept at (0,0).

  • Passes through the origin and bisects the first and third quadrants at a 45-degree angle with the x-axis.

• Quadratic Function: $f(x) = x^2$

  • Graph shape: U-shape, called a parabola.

• Cubic Function: $f(x) = x^3$

  • Graph is the cube function.

• Square Root Function: $f(x) = ext{sqrt}(x)$

  • Graph resembles one branch of a parabola turned on its side.

• Cube Root Function: $f(x) = ext{cbrt}(x)$

• Absolute Value Function: $f(x) = |x|$

  • Graph is a V-shape with a point at the origin.

• Reciprocal Function: $f(x) = rac{1}{x}$

  • Graph features two branches that approach the x-axis (horizontal asymptote) without touching it.

  • For large values of x and as x approaches negative infinity, the branches tend toward the x-axis.

  • For x close to zero, the function approaches the y-axis (vertical asymptote) without touching it.

Understanding the Parent Function

• For example, $f(x) = |x| + 1$ involves shifting the parent function $|x|$ upward by 1 unit.

• Demonstration of how changing the function affects its graph:

  • For example, using a table of ordered pairs, we can elaborate that when:

  • $f(-3) = | -3 | + 1 = 4$

  • $f(-2) = | -2 | + 1 = 3$

  • $f(0) = | 0 | + 1 = 1$

  • Showing that y-values of the function $f$ are consistently one unit more than the parent function.

Transformations of Functions

• Vertical Shifts:

  • If $k$ is a positive real number, then:

  • The graph of $y = f(x) + k$ shifts up by k units.

  • The graph of $y = f(x) - k$ shifts down by k units.

• Horizontal Shifts:

  • If $h$ is a positive real number, then:

  • The graph of $y = f(x - h)$ shifts to the right by h units.

  • The graph of $y = f(x + h)$ shifts to the left by h units.

Specific Function Examples

• For $g(x) = ext{sqrt}(x) + 5$:

  • This indicates a vertical shift upwards by 5 units while keeping the x-values unchanged if evaluated.

• For $g(x) = ext{sqrt}(x + 5)$:

  • This indicates a horizontal shift left by 5 units.

  • The domain changes from $[0, ext{inf})$ to $[-5, ext{inf})$.

  • Example of finding the x-intercept for $g$:

  • Setting $g(x) = 0$ gives:

    • $0 = ext{sqrt}(x + 5)$

    • Squaring both sides results in $x + 5 = 0 <br>ightarrow x = -5$

  • The x-intercept is (-5, 0).

Piecewise Function Introduction

• Introducing a piecewise defined function using a graphing calculator to visualize the function's behavior.

Greatest Integer Function

• Defined as: $f(x) = ext{int}(x)$

  • It returns the largest integer less than or equal to x.

  • Example calculations:

  • For x = 1: $f(1) = 1$

  • For x = 1.2: $f(1.2) = 1$

  • For x = 1.4: $f(1.4) = 1$

Even and Odd Functions

• Analyzing if functions are even, odd, or neither:

  • For $k(x) = 13x^3 + 12x$:

  • Finding $k(-x)$ produces the same function, indicating it is an odd function.

  • If $z(x) = ext{sqrt}(49 + x^2)$:

  • Since $z(-x) = z(x)$, it confirms that this function is even.

Conclusion

• Recap of the functions discussed, emphasizing understanding transformations, definitions, and characteristics of each basic function graph.