- Identify the slope \(m = -5\) from the given linear function \(f(x) = -5x + 2\).
- Recognize the given point \((x_1, y_1) = (-2, 12)\).
- Use the point-slope formula \(y - y_1 = m(x - x_1)\) to substitute the identified values.
- Resulting equation is \(y - 12 = -5(x + 2)\), which corresponds to the first option.
- Final answer: \(\boxed{y - 12 = -5(x + 2)}\)
Explanation
- Analyze the Given Function and Point
The function provided is \(f(x) = -5x + 2\). This is in the slope-intercept form \(y = mx + b\), where the slope \(m\) is the coefficient of \(x\), which is \(-5\). We are given that the line contains the point \((-2, 12)\). This means \(x_1 = -2\) and \(y_1 = 12\). We need to express this line in the point-slope form.
- Apply the Point-Slope Form Formula
The point-slope form of a linear equation is defined by the formula:
\[y - y_1 = m(x - x_1)\]
where \(m\) is the slope and \((x_1, y_1)\) is a point on the line. Substituting our values, where \(m = -5\), \(x_1 = -2\), and \(y_1 = 12\), we get:
\[y - 12 = -5(x - (-2))\]
Simplifying the expression within the parentheses, \((x - (-2))\) becomes \((x + 2)\). Therefore, the point-slope form is:
\[y - 12 = -5(x + 2)\]
- Select the Correct Option
Comparing our result \(y - 12 = -5(x + 2)\) with the given options, we identify that this expression is the first choice provided. The other options involve incorrect slopes or incorrect signs related to the point coordinates. Thus, the correct equation is \(y - 12 = -5(x + 2)\).
Examples
In economics, the point-slope form is used to model how the quantity demanded of a product changes as its price changes. If a company knows their product's demand curve is linear and they observe a specific market condition where at a price of 12 dollars they sell -2 units relative to a baseline (or shift), they can use the point-slope form to determine the entire demand function. This allows them to predict sales at different price points and optimize their pricing strategy.