The graph of a line in the x y -plane passes through the...

The graph of a line in the $x y$ -plane passes through the point $(-2, k)$ and crosses the $x$ -axis at the point $(-4,0)$ . The line crosses the $y$ -axis at the point $(0,12) .$ What is the value of $k ?$

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Key Concepts

Intercepts

Intercepts refer to the points where a line crosses the coordinate axes. The x-intercept is the point where the line meets the x-axis (y = 0) and the y-intercept is where the line meets the y-axis (x = 0). Understanding these points is crucial for constructing the equation of a line and for visualizing its position in the plane.

Slope

The slope of a line represents its steepness and direction, calculated as the ratio of the change in y to the change in x between any two points on the line. It is a fundamental concept in coordinate geometry and is essential for describing the behavior and equation of a line.

Slope-Intercept Form

The slope-intercept form of a line, given as y = mx + b, is a way to express the equation of a line using its slope (m) and y-intercept (b). This form allows for quick graphing and understanding of how the line crosses the y-axis, as well as how it behaves as x changes.

Using Known Points to Determine Unknowns

When the equation of a line is partly known, known points (such as intercepts or additional coordinates) can be substituted into the equation to solve for unknown parameters. This technique is fundamental in algebra and coordinate geometry to complete the description of a line.

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