Algebra Review: Zeros, Roots, and X-Intercepts

Key Concepts

• Zeros, Roots, and X-Intercepts are different terms used interchangeably to denote points where the function equals zero.
• Zeros of a function: Values of x for which f(x) = 0.
• Roots of an equation: Solutions to the equation when set to zero.
• X-Intercepts: Points where the graph crosses the x-axis.

Example Function

• Function: f(x) = x² - 1
• Objective: Find zeros of the function, meaning solve x² - 1 = 0.

Solving the Equation

Method 1: Solving Directly

1. Add 1 to both sides: x² = 1
2. Take the square root: x = ±1

Method 2: Factoring

1. Recognize the difference of squares: x² - 1 = (x - 1)(x + 1)
2. Set each factor to zero:
   • x - 1 = 0 → x = 1
   • x + 1 = 0 → x = -1

• Solution: Zeros are x = ±1
• Graph: A parabola with vertex at (0, -1), opening upwards with x-intercepts at (1,0) and (-1,0).

Graphing X-Intercepts

• The x-intercepts are points where the graph of the function crosses x-axis.
• For f(x) = x² - 1, these are (1, 0) and (-1, 0).

Understanding Equations and Zeros

• When given an equation, e.g., x² - 2x - sin(x) = 0, find the expression that equals zero.
• Example: Subtract sin(x) to isolate the expression x² - 2x - sin(x) = 0.
• The function is f(x) = x² - 2x - sin(x).

Subtracting to Find Functions

1. Subtract terms to isolate expressions equal to zero.
2. Remember different forms may represent the same zeros.

Multiple Ways to Reach the Same Result

• Different manipulations of the equation may yield the same zeros.
• Example: Using subtracting or adding different terms on both sides can lead you to the same function.

Practice Example

• Equation: 0 = e^x + 4x² - 3x + 5
• Function: f(x) = e^x + 4x² - 3x + 5
• Alternative Solution: 0 = 3x - 5 - e^x - 4x²
• Both forms have the same zeros.

Key Takeaways

• Zeros are crucial in graphing functions and understanding their behavior.
• The expression set equal to zero in an equation is the function whose zeros you are finding.
• Focus on understanding how to set equations to zero and identify corresponding functions.