Graphing Y = -4|x + 5| A Step-by-Step Guide

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In this comprehensive guide, we will delve into the process of graphing the absolute value equation y = -4|x + 5|. Absolute value equations often present a unique challenge due to the nature of the absolute value function, which introduces a 'V' shape into the graph. Understanding the transformations applied to the parent function, y = |x|, is crucial for accurately plotting the given equation. We will explore the effects of the coefficients and constants within the equation, including vertical stretches, reflections, and horizontal translations. By breaking down the equation into its components and analyzing their individual impacts on the graph, we can systematically construct the graph and identify key features such as the vertex, axis of symmetry, and the overall orientation of the 'V' shape.

Understanding the Parent Function: y = |x|

Before we tackle the given equation, let's establish a firm understanding of the parent function, y = |x|. This function forms the basis for all absolute value graphs, and its characteristics serve as a reference point for analyzing transformations. The absolute value of a number is its distance from zero, which means |x| is always non-negative. This results in a 'V' shaped graph with the vertex at the origin (0, 0). The graph consists of two linear segments: one with a slope of 1 for x ≥ 0 and another with a slope of -1 for x < 0. The symmetry of the graph is evident, with the y-axis acting as the axis of symmetry. To plot the parent function, we can select a few key points:

  • When x = -2, y = |-2| = 2
  • When x = -1, y = |-1| = 1
  • When x = 0, y = |0| = 0 (the vertex)
  • When x = 1, y = |1| = 1
  • When x = 2, y = |2| = 2

Plotting these points and connecting them, we obtain the characteristic 'V' shape of the absolute value function. This understanding of the parent function is crucial as we move on to analyzing the transformations applied in the equation y = -4|x + 5|.

Identifying Transformations

The equation y = -4|x + 5| represents a transformation of the parent function y = |x|. To graph this equation effectively, we must first identify the transformations that have been applied. By comparing the given equation to the standard form of an absolute value equation, y = a|x - h| + k, we can readily discern these transformations.

  1. Vertical Stretch/Compression and Reflection: The coefficient a in the equation y = a|x - h| + k determines the vertical stretch or compression and any reflection across the x-axis. In our equation, y = -4|x + 5|, a = -4. The negative sign indicates a reflection across the x-axis, meaning the 'V' shape will be inverted. The absolute value of a, which is 4, indicates a vertical stretch by a factor of 4. This means the graph will be steeper than the parent function.

  2. Horizontal Translation: The term (x - h) inside the absolute value represents a horizontal translation. In our equation, we have (x + 5), which can be rewritten as (x - (-5)). This means h = -5, indicating a horizontal translation of 5 units to the left. The vertex of the graph will be shifted 5 units to the left from the origin.

By identifying these transformations, we have a clear roadmap for graphing the equation. We know the graph will be a 'V' shape, reflected across the x-axis, vertically stretched by a factor of 4, and translated 5 units to the left.

Determining the Vertex

The vertex is a critical point in the graph of an absolute value equation. It is the point where the two linear segments of the 'V' shape meet, and it represents either the minimum or maximum value of the function. In the standard form of an absolute value equation, y = a|x - h| + k, the vertex is located at the point (h, k). In our equation, y = -4|x + 5|, we can rewrite it as y = -4|x - (-5)| + 0. Comparing this to the standard form, we can identify h = -5 and k = 0.

Therefore, the vertex of the graph is at the point (-5, 0). This point will serve as the cornerstone for plotting the graph. We know the 'V' shape will originate from this point, opening downwards due to the reflection across the x-axis.

Finding Additional Points

To accurately graph the equation, we need to determine additional points on either side of the vertex. These points will help us define the slope and shape of the 'V'. Since the graph is symmetrical about the vertical line passing through the vertex, we can choose x-values that are equidistant from the x-coordinate of the vertex. Let's select the following x-values:

  • x = -6 (1 unit to the left of the vertex)
  • x = -4 (1 unit to the right of the vertex)
  • x = -7 (2 units to the left of the vertex)
  • x = -3 (2 units to the right of the vertex)

Now, we will substitute these x-values into the equation y = -4|x + 5| and calculate the corresponding y-values:

  • When x = -6, y = -4|-6 + 5| = -4|-1| = -4
  • When x = -4, y = -4|-4 + 5| = -4|1| = -4
  • When x = -7, y = -4|-7 + 5| = -4|-2| = -8
  • When x = -3, y = -4|-3 + 5| = -4|2| = -8

This gives us the following points: (-6, -4), (-4, -4), (-7, -8), and (-3, -8). These points, along with the vertex (-5, 0), provide sufficient information to plot the graph.

Plotting the Graph

Now that we have identified the vertex and additional points, we can proceed to plot the graph of y = -4|x + 5|. First, plot the vertex at (-5, 0) on the coordinate plane. This is the starting point of our 'V' shape. Next, plot the additional points we calculated: (-6, -4), (-4, -4), (-7, -8), and (-3, -8). These points define the arms of the 'V'.

Connect the vertex to the points on either side with straight lines. Due to the reflection across the x-axis, the 'V' shape will open downwards. The vertical stretch of 4 makes the graph steeper than the parent function. The resulting graph is a 'V' shape with its vertex at (-5, 0), opening downwards, and stretched vertically. The axis of symmetry is the vertical line x = -5.

Key Features of the Graph

Once we have plotted the graph, it's important to identify and summarize its key features. These features provide a comprehensive understanding of the function's behavior.

  1. Vertex: As we determined earlier, the vertex of the graph is at (-5, 0). This is the maximum point on the graph, as the 'V' shape opens downwards.

  2. Axis of Symmetry: The axis of symmetry is the vertical line that passes through the vertex, dividing the graph into two symmetrical halves. In this case, the axis of symmetry is the line x = -5.

  3. Domain: The domain of an absolute value function is all real numbers, as there are no restrictions on the x-values that can be input into the function. Therefore, the domain of y = -4|x + 5| is (-∞, ∞).

  4. Range: The range of the function is the set of all possible y-values. Since the graph opens downwards and the vertex is the maximum point, the range includes all y-values less than or equal to the y-coordinate of the vertex. Therefore, the range of y = -4|x + 5| is (-∞, 0].

  5. Intercepts:

    • x-intercept: The x-intercept is the point where the graph intersects the x-axis. In this case, the vertex (-5, 0) is also the x-intercept.
    • y-intercept: The y-intercept is the point where the graph intersects the y-axis. To find the y-intercept, we set x = 0 in the equation and solve for y: y = -4|0 + 5| = -4|5| = -20. Therefore, the y-intercept is (0, -20).

Conclusion

Graphing the absolute value equation y = -4|x + 5| involves understanding the transformations applied to the parent function y = |x|. By identifying the vertical stretch, reflection, and horizontal translation, we can systematically plot the graph. Determining the vertex and additional points is crucial for accurately depicting the shape and orientation of the 'V'. Finally, identifying key features such as the axis of symmetry, domain, range, and intercepts provides a comprehensive understanding of the function's behavior. This step-by-step approach can be applied to graphing any absolute value equation, making it a valuable tool in mathematical analysis.