Problem 3 - Calculus Single Variable

Question

Sketch, on the same set of axes, lines having slope -3 and passing through points \((-2,-7),(-1,0),(3,0),\) and (5,1).

Step-by-Step Solution

Answer

Plot four lines with equations: 1)\(y=-3x-1\), 2) \(y=-3x-3\), 3) \(y=-3x+9\), 4) \(y=-3x+16\).

Step 1: Understand the Problem

You need to sketch lines on a set of axes that have a slope of -3 and pass through specific points: $(-2, -7)$, $(-1, 0)$, $(3, 0)$, and $(5, 1)$. The slope-means the lines will slant downwards from left to right.

Step 2: Write Equation for a Line With Given Slope Through a Point

To sketch a line with a given slope through a point, use the point-slope form of the line equation: \( y - y_1 = m(x - x_1) \), where \( m = -3 \) is the slope, and \((x_1, y_1)\) is the given point.

Step 3: Line Through Each Point

For each point, substitute into the linear equation:1. **For** \((-2, -7)\): \( y + 7 = -3(x + 2) \on simplifying y = -3x - 1 \) 2. **For** \((-1, 0)\): \( y = -3(x + 1) = -3x - 3\) 3. **For** \((3, 0)\): \( y = -3(x - 3) = -3x + 9\)4. **For** \((5, 1)\): \( y - 1 = -3(x - 5) = -3x + 16\) These equations represent the lines to be sketched.

Step 4: Plot the Lines

For each equation, plot the corresponding line:- **Line 1:** Starts at \((-2, -7)\) with equation \(y = -3x - 1\)- **Line 2:** Starts at \((-1, 0)\) with equation \(y = -3x - 3 \)- **Line 3:** Starts at \((3, 0)\) with equation \(y = -3x + 9\)- **Line 4:** Starts at \((5, 1)\) with equation \(y = -3x + 16\)Use graph paper to ensure accuracy. Each line is determined by its slope and intercepts from the equation.

Key Concepts

Linear Equations

Linear equations are fundamental in mathematics, known for their simplicity and predictability. They represent straight lines when graphed on a coordinate plane. These equations have a standard form that is easy to recognize: \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
Linear equations can be solved using various methods, including substitution, elimination, and graphing. The primary goal is to find the value of \(y\) for a given \(x\), or vice versa. This kind of equation relates two variables, implying that a change in one variable induces a change in the other, consistent with the equation's slope and intercept.

• The slope \(m\) indicates how steep the line is.
• The y-intercept \(b\) is where the line crosses the y-axis.

Understanding linear equations allows you to predict and interpret relationships between quantities that change at a constant rate.

Slope

Slope is a crucial concept in understanding how lines behave on a graph. It measures how steep a line is and its direction. Mathematically, slope is represented as \(m\) in the equation \(y = mx + b\) and is calculated as the change in the vertical direction divided by the change in the horizontal direction, \(\frac{\Delta y}{\Delta x}\).
The slope of a line determines its angle and direction:

• A positive slope means the line rises upwards as it moves from left to right.
• A negative slope indicates that the line falls downwards to the right.
• A zero slope results in a horizontal line, meaning no vertical change.
• An undefined slope, where \(\Delta x = 0\), results in a vertical line.

Understanding the slope helps to visualize how quickly one variable changes in relation to another and is essential in fields like physics, economics, and statistics.

Graphing Lines

Graphing lines on a coordinate plane is a vital skill for visualizing linear equations. This process transforms abstract algebraic equations into tangible geometric figures. By using the point-slope form, \(y - y_1 = m(x - x_1)\), you can easily identify where to start plotting the line and how it proceeds across the graph.
Each line is defined by its equation and can be plotted by:

• Identifying a point \,\((x_1, y_1)\)\, on the line.
• Using the slope \(m\) to determine the line's direction.
• Drawing a straight line through the initial point that follows the slope.

The goal of graphing is to provide a visual perspective of relationships between variables, making abstract concepts more accessible. Whether it is for comparison, analysis, or solution finding, graphical representation simplifies understanding complex linear relations.