-10 -8 -6 -4 -2 0 2 4 6 8 10 10 y 8 6 4 2 0 -2 -4 -6 -8 -10 x
The Correct Answer and Explanation is:
The graph shows a straight line passing through the coordinate plane. By observing its key intercepts, we can determine its equation.
The line crosses the y-axis at the point (0, 2) and the x-axis at approximately (–2, 0). From these two points, we can calculate the slope mm using the slope formula:
m=y2−y1x2−x1=0−2−2−0=−2−2=1m = \frac{y_2 – y_1}{x_2 – x_1} = \frac{0 – 2}{-2 – 0} = \frac{-2}{-2} = 1
Now that we know the slope is 1 and the y-intercept is 2, we can write the equation of the line in slope-intercept form y=mx+by = mx + b:
y=x+2y = x + 2
Explanation
The equation y=x+2y = x + 2 represents a linear function, one whose graph is a straight line. Every linear function has a constant rate of change, known as its slope. Here, the slope is 1, which means that for each unit increase in x, the y-value increases by the same amount. This gives the line a consistent rise over run of 1, making it ascend diagonally from left to right at a 45-degree angle.
The y-intercept, which is the point where the line crosses the vertical y-axis, occurs at (0, 2). This tells us that when no input (x = 0) is applied to the function, the output (y) is 2. The x-intercept, where the line crosses the horizontal x-axis, is at approximately (–2, 0). This occurs when the output value y is zero, giving us a helpful verification of the equation since plugging –2 into the equation yields y=–2+2=0y = –2 + 2 = 0.
This linear graph divides the coordinate plane into two regions. Points above the line satisfy the inequality y>x+2y > x + 2, while points below satisfy y<x+2y < x + 2. Because the slope is positive, the function is increasing, meaning the function outputs grow as the input increases.
The consistent pattern, symmetry, and predictable growth make this line a clear example of a linear relationship in algebra.
