Problem 50 - Algebra 2

Question

Graph the line with the given equation. \(y+2=-2(x-1)\)

Step-by-Step Solution

Answer

The line is graphed with a slope of \(-2\) and a y-intercept of 0, passing through points (0,0) and (1,-2).

Step 1: Simplify the Equation

Start by simplifying the given equation into the slope-intercept form \(y = mx + b\). We have:\[ y + 2 = -2(x - 1) \]First, distribute the \(-2\):\[ y + 2 = -2x + 2 \]Next, subtract \(2\) from both sides to isolate \(y\):\[ y = -2x + 2 - 2 \]\[ y = -2x \]The equation of the line is now in the form \(y = -2x\).

Step 2: Identify Slope and Y-Intercept

From the simplified equation, \(y = -2x\), we can identify that the slope \(m\) is \(-2\) and the y-intercept \(b\) is \(0\). This means the line crosses the y-axis at the origin (0,0).

Step 3: Plot the Y-Intercept

Begin plotting the graph by drawing a point at the y-intercept. In this case, the y-intercept is at the origin (0,0).

Step 4: Use the Slope to Find Another Point

Using the slope \(-2\), which means that for every 1 unit you move to the right (positive x-direction), you move 2 units down (negative y-direction), find another point. From the origin (0,0), move 1 unit right to (1,0) and then 2 units down to (1,-2). Plot this point.

Step 5: Draw the Line

Use a ruler to draw a straight line through the points (0,0) and (1,-2). This line is the graph of the equation \(y = -2x\).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way to write the equation of a straight line. It's one of the most common forms and is handy for graphing linear equations.
The general formula for the slope-intercept form is\[ y = mx + b \] where:

• \(m\) is the slope of the line
• \(b\) is the y-intercept, which is the point where the line crosses the y-axis.

This form makes it easy to identify the slope and "b" value just by looking at the equation. Once you have these values, plotting the line becomes straightforward.

Slope

The slope of a line is a number that describes both the direction and the steepness of the line. In the equation \(y = mx + b\), the slope is represented by the letter \(m\).

• If the slope is positive, the line goes uphill (from left to right).
• If the slope is negative, the line goes downhill.
• A larger absolute value of the slope indicates a steeper line.
• Zero slope means the line is horizontal.

Understanding slope is crucial for graphing because it tells you how to move from one point to another on the line. For instance, if we have a slope of \(-2\), it indicates that for every 1 unit you move to the right, you move 2 units down.

Y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis. In the slope-intercept form \(y = mx + b\), the \(b\) value represents the y-intercept.

• If \(b = 0\), the line crosses the origin (0,0).
• The y-intercept is crucial for starting the graph, as it gives you the initial point on the graph.
• It shows where the line cuts through the y-axis when the value of \(x\) is zero.

Finding the y-intercept is the first step in graphing a line, as it helps set the stage for placing additional points by using the slope.

Plotting Points

Plotting points is the final step in graphing a linear equation once you have identified the slope and y-intercept. Starting with the y-intercept, you mark it on the y-axis, which gives you your first point.

• Utilize the slope to determine your next point. For example, with a slope of \(-2\), begin at the y-intercept and move horizontally (right if positive direction, left if negative) one unit, then vertically (up if positive slope, down if negative) to locate a second point.
• Once you have two points, you can draw a line through them to complete the graph.

These points, derived from the slope and y-intercept, are sufficient to define a line. Graphing becomes a simple matter of connecting these plotted points with a straight line.