Problem 13 - Prealgebra

Question

Complete the given ordered pairs, and use the results to graph the equation. (GRAPH CANT COPY) $$y=\frac{1}{2} x+2 \quad(-2,),(0,),(2,)$$

Step-by-Step Solution

Answer

The completed pairs are \((-2, 1), (0, 2), (2, 3)\). Plot these to graph the line.

Step 1: Identifying the Problem

We are given an equation of a line, \( y = \frac{1}{2} x + 2 \), and we need to complete the ordered pairs \((-2, ), (0, ), (2, )\) and graph the line using these points.

Step 2: Calculating the Y-coordinate for x = -2

Substitute \( x = -2 \) into the equation to find \( y \). We have: \[ y = \frac{1}{2}(-2) + 2 = -1 + 2 = 1 \]. So the complete ordered pair is \((-2, 1)\).

Step 3: Calculating the Y-coordinate for x = 0

Substitute \( x = 0 \) into the equation to find \( y \). We have: \[ y = \frac{1}{2}(0) + 2 = 0 + 2 = 2 \]. So the complete ordered pair is \((0, 2)\).

Step 4: Calculating the Y-coordinate for x = 2

Substitute \( x = 2 \) into the equation to find \( y \). We have: \[ y = \frac{1}{2}(2) + 2 = 1 + 2 = 3 \]. So the complete ordered pair is \((2, 3)\).

Step 5: Graphing the Equation

With the complete ordered pairs \((-2, 1), (0, 2), (2, 3)\), plot these points on a graph. Each of these points lies on a straight line, validating our equation \( y = \frac{1}{2}x + 2 \). Draw a line through these points to represent the equation.

Key Concepts

Understanding Ordered Pairs

Ordered pairs are a fundamental concept in graphing linear equations and plotting on a coordinate plane. They consist of two elements arranged in a specific sequence. The first element is typically the x-coordinate, and the second element is the y-coordinate. Ordered pairs are always written in the form

• \((x, y)\)

The x-coordinate indicates how far a point is along the x-axis (horizontal), and the y-coordinate shows how far the point is along the y-axis (vertical). Together, these coordinates help to precisely locate any point on a plane. For instance, in the ordered pairs given

• \((-2, 1)\)
• \((0, 2)\)
• \((2, 3)\)

These values guide us to plot the points directly on a graph by moving horizontally or vertically based on the x and y values.

Decoding the Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most common ways to express a line. It is represented as \[ y = mx + b \] where:

• \(m\) is the slope of the line
• \(b\) is the y-intercept

The slope \(m\) indicates the steepness of the line. It tells us how much the y-coordinate changes for a unit change in the x-coordinate. For example, if the slope is \(\frac{1}{2}\), for every increase of 1 unit in x, y increases by half of that – 0.5 units. Meanwhile, the y-intercept \(b\) is the point where the line crosses the y-axis. In our exercise's equation, \(b\) is 2, meaning the line will cross the y-axis at the point \((0, 2)\). This form not only allows easy graphing but also offers immediate insight into the line's behavior.

Plotting Points on a Graph

Plotting points is an essential skill for visualizing equations and their solutions. To plot a point on the Cartesian coordinate system, start by identifying its ordered pair, \((x, y)\). Follow these steps:

• Begin at the origin, where the x and y axes intersect.
• Move along the x-axis by the x-coordinate value. Positive values go to the right, and negative values go to the left.
• From that position, move parallel to the y-axis by the y-coordinate value. Positive values go up, and negative go down.

Mark the point where you finish. For the exercise, using \((-2, 1)\), Follow this process: Move 2 units left from the origin and then 1 unit up to locate the point. Repeat similarly for the other pairs \((0, 2)\) and \((2, 3)\). Once all points are plotted, verify these exist on a straight line, which confirms the equation's accuracy. Draw a line through these points, showing visually that they are part of the line described by \(y = \frac{1}{2}x + 2\).