For each equation, (a) give a table with at least three ordered pairs that are solutions, and (b) graph the equation. See Examples 7 and 8. y=|x+4|

Verified step by step guidance

1. Step 1: Understand the equation y = |x + 4|. This is an absolute value function, which means it will create a V-shaped graph. The vertex of this graph is at the point where the expression inside the absolute value is zero, i.e., x + 4 = 0, which gives x = -4.
2. Step 2: Create a table of values. Choose values for x around the vertex to find corresponding y values. For example, select x = -6, x = -4, and x = -2.
3. Step 3: Calculate the y values for each chosen x value. For x = -6, y = |-6 + 4| = 2. For x = -4, y = |-4 + 4| = 0. For x = -2, y = |-2 + 4| = 2.
4. Step 4: Fill in the table with the ordered pairs: (-6, 2), (-4, 0), and (-2, 2). These are points on the graph of the equation.
5. Step 5: Plot these points on a coordinate plane and draw the V-shaped graph. The graph will have a vertex at (-4, 0) and will open upwards, with the arms of the V extending symmetrically from the vertex.

Key Concepts

Absolute Value Function

The absolute value function, denoted as |x|, measures the distance of a number x from zero on the number line, always yielding a non-negative result. In the equation y = |x + 4|, the expression inside the absolute value, x + 4, shifts the graph horizontally to the left by 4 units. Understanding this function is crucial for generating ordered pairs and accurately graphing the equation.

Ordered Pairs

Ordered pairs are pairs of numbers used to represent points in a coordinate system, typically written as (x, y). For the equation y = |x + 4|, finding ordered pairs involves substituting different x-values into the equation to calculate corresponding y-values. This process is essential for creating a table of solutions that visually represents the relationship defined by the equation.

Graphing Linear and Non-linear Equations

Graphing involves plotting points on a coordinate plane to visualize the relationship between variables. For the equation y = |x + 4|, the graph will form a 'V' shape, indicating that the function is non-linear due to the absolute value. Understanding how to graph such equations helps in interpreting their behavior and identifying key features like intercepts and vertex.