The graph of a linear polynomial is a straight line.
A linear polynomial is an expression of the form ax + b, where a and b are constants, and x is a variable. When we graph this polynomial, we are essentially plotting all the points (x, y) that satisfy the equation y = ax + b. Because the highest power of x is 1, the resulting graph will always be a straight line.
Key Characteristics:
- Straight Line: The graph is always a straight line, extending infinitely in both directions.
- X-intercept: The line will intersect the x-axis at one point, which represents the root or zero of the polynomial (i.e., the value of x for which ax + b = 0).
- Y-intercept: The line will intersect the y-axis at one point, which is the value of b in the equation y = ax + b.
- Slope: The constant a represents the slope of the line, indicating its steepness and direction. A positive slope means the line rises from left to right, while a negative slope means it falls from left to right. A slope of zero signifies a horizontal line.
Example:
Consider the linear polynomial y = 2x + 1.
- The graph of this equation will be a straight line.
- To plot it, you can find at least two points that satisfy the equation. For example:
- If x = 0, then y = 2(0) + 1 = 1. So, the point (0, 1) lies on the line (y-intercept).
- If x = 1, then y = 2(1) + 1 = 3. So, the point (1, 3) lies on the line.
- By connecting these two points, you can draw the entire straight line representing the graph of the linear polynomial y = 2x + 1.
In summary, the graph of any linear polynomial of the form ax + b will always be a straight line, characterized by its slope and intercepts.
