Which graph represents the function $y-3=\frac{3}{2}(x-4)$?
The given equation is \(y - 3 = \frac{3}{2}(x - 4)\). This is in the point-slope form \(y - y_1 = m(x - x_1)\), where \(m\) is the slope of the line and \((x_1, y_1)\) is a point on the line. By comparing the given equation to this form, we can identify that the slope is \(m = \frac{3}{2}\) and that the line passes through the point \((4, 3)\). To better visualize and identify the line, we can convert it into the slope-intercept form \(y = mx + b\).
To convert the equation to slope-intercept form, we start with \(y - 3 = \frac{3}{2}(x - 4)\). Distributing the slope \(\frac{3}{2}\) gives \(y - 3 = \frac{3}{2}x - \frac{3}{2} \times 4\), which simplifies to \(y - 3 = \frac{3}{2}x - 6\). Adding 3 to both sides, we get \(y = \frac{3}{2}x - 3\). The y-intercept is \((0, -3)\). We now have the slope \(m = \frac{3}{2}\) and the y-intercept \(-3\). A line with these properties passes through \((0, -3)\) and \((4, 3)\). Any graph representing this function must pass through these two points.
Understanding how to identify a line's properties from its equation is fundamental for graphing linear functions in fields like physics or economics. For example, if you are modeling the cost of producing items where the total cost depends on a fixed base cost and a variable cost per item, the equation \(y = mx + b\) helps you visualize the relationship, where the slope \(m\) represents the cost per unit and the y-intercept \(b\) represents the initial setup cost. Mastery of these forms helps in quickly interpreting real-world data and trends.