Finding Point-Slope Form Mr Shaw Linear Function Graph
What is the point-slope form of the equation of the line given that the line contains the point (-2,12) and the function is f(x) = -5x + 2?
This article delves into the process of determining the point-slope form of a linear equation, using the example of Mr. Shaw graphing the function f(x) = -5x + 2. We'll explore the concepts behind linear equations, point-slope form, and how to apply them to solve this specific problem. Understanding these concepts is crucial for anyone studying algebra and beyond, as linear functions form the foundation for more advanced mathematical topics.
Understanding Linear Functions
In linear functions, the relationship between two variables, typically denoted as x and y, is represented by a straight line on a graph. The most common way to represent a linear function is the slope-intercept form, y = mx + b, where m represents the slope of the line and b represents the y-intercept (the point where the line crosses the y-axis). However, the point-slope form offers another valuable perspective, particularly when you know a specific point on the line and its slope.
The slope, m, quantifies the steepness and direction of the line. A positive slope indicates that the line rises as you move from left to right, while a negative slope indicates that it falls. The magnitude of the slope represents how rapidly the line rises or falls; a larger magnitude signifies a steeper line. The y-intercept, b, is the y-coordinate of the point where the line intersects the y-axis, providing a fixed point of reference for the line's position.
Linear functions are prevalent in various real-world scenarios. For example, they can model the relationship between time and distance traveled at a constant speed, the cost of a product based on the number purchased, or the depreciation of an asset over time. Their simplicity and predictability make them a fundamental tool in mathematical modeling and analysis.
Point-Slope Form: An Alternative Representation
The point-slope form of a linear equation is expressed as y - y₁ = m(x - x₁), where m is the slope of the line and (x₁, y₁) is a known point on the line. This form is particularly useful when you have a point and the slope and want to write the equation of the line. It directly incorporates the slope and a specific point, making it easy to visualize and understand the line's characteristics.
The point-slope form highlights the relationship between any point (x, y) on the line and the known point (x₁, y₁). The term (y - y₁) represents the vertical change, and (x - x₁) represents the horizontal change. The slope, m, then represents the ratio of these changes, ensuring that any point on the line satisfies the given slope and passes through the known point.
Converting between slope-intercept form and point-slope form is a common algebraic exercise. To convert from point-slope form to slope-intercept form, you simply distribute the m and solve for y. Conversely, to convert from slope-intercept form to point-slope form, you need to identify a point on the line and substitute its coordinates into the point-slope form equation.
Analyzing Mr. Shaw's Function: f(x) = -5x + 2
Mr. Shaw graphs the function f(x) = -5x + 2. This equation is already in slope-intercept form, y = mx + b, where m = -5 and b = 2. This tells us that the line has a slope of -5, meaning it descends as you move from left to right, and the y-intercept is 2, meaning it crosses the y-axis at the point (0, 2). The equation f(x) = -5x + 2 is a concise way to represent the relationship between x and y for all points on this line.
The slope of -5 indicates that for every 1 unit increase in x, the value of y decreases by 5 units. This steep negative slope makes the line descend rapidly. The y-intercept of 2 provides a fixed reference point on the line, ensuring its precise location on the coordinate plane. Understanding the slope and y-intercept provides a complete picture of the line's behavior.
Furthermore, the problem states that the line contains the point (-2, 12). This information is crucial because we can use it, along with the slope, to write the equation of the line in point-slope form. The point (-2, 12) means that when x = -2, y = 12. This specific point satisfies the equation f(x) = -5x + 2, and we'll use it to transition into point-slope form.
Determining the Point-Slope Form
The point-slope form of a linear equation is y - y₁ = m(x - x₁). We are given the slope, m = -5, and a point on the line, (-2, 12). We can substitute these values into the point-slope form equation to find the equation of the line Mr. Shaw graphed.
Substituting m = -5, x₁ = -2, and y₁ = 12 into the point-slope form, we get: y - 12 = -5(x - (-2)). Simplifying the expression inside the parentheses, we have y - 12 = -5(x + 2). This is the point-slope form of the equation of the line.
The equation y - 12 = -5(x + 2) represents the same line as f(x) = -5x + 2, but it emphasizes a specific point on the line and its slope. It directly shows that the line passes through the point (-2, 12) and has a slope of -5. This point-slope form is particularly useful in certain applications where knowing a specific point and the slope is more insightful than the y-intercept.
Why the Point-Slope Form is the Answer
The correct answer is y - 12 = -5(x + 2). This equation precisely captures the information given in the problem: the slope of the line is -5, and the line passes through the point (-2, 12). The point-slope form directly reflects these pieces of information, making it the most intuitive representation in this context.
Other options might be close, but they would either have the wrong slope or use the point (-2, 12) incorrectly. For example, an equation with a slope other than -5 would represent a different line altogether. Similarly, an equation using the point (-2, 12) with the incorrect sign (e.g., y + 12 = -5(x - 2)) would represent a line passing through a different point.
The point-slope form is valuable because it allows us to write the equation of a line directly from a point and its slope. This is particularly useful in problems where the y-intercept is not explicitly given or when the focus is on the behavior of the line around a specific point.
Conclusion
In conclusion, by understanding the point-slope form of a linear equation and how to apply it, we can effectively solve problems like this one. Mr. Shaw's function, f(x) = -5x + 2, which contains the point (-2, 12), can be expressed in point-slope form as y - 12 = -5(x + 2). This exercise highlights the versatility and importance of different forms of linear equations in mathematical problem-solving.
The point-slope form provides a powerful tool for understanding and representing linear relationships. By grasping the concepts behind slope, points on a line, and the different forms of linear equations, you can confidently tackle a wide range of algebraic problems and applications.