Graphing Absolute Value and Square Root Functions

1. Graphing Absolute Value Functions

An absolute value function involves the absolute value of a variable. Remember, the absolute value of a number is its distance from zero on the number line, which is always non-negative. For example, |5| = 5 and |-5| = 5.

Mathematically, the parent absolute value function f(x) = |x| is defined as a piecewise function:
$f(x)=|x|=\{\begin{array}{ll}x& \text{if }x\ge 0\\ -x& \text{if }x<0\end{array}$
This definition is what creates the characteristic "V" shape.

To understand the parent graph and how we can apply transformations to it, let's watch the first part of a helpful video.

Transformations of Absolute Value Functions

Just as we transformed y = x² into y = a(x - h)² + k, we can transform the parent function y = |x| into the general form:
$f(x)=a|x-h|+k$

The parameters a, h, and k have the same effects you're already familiar with:

• The vertex (the sharp point of the "V") is located at (h, k).
• The parameter h controls the horizontal translation (shift). Remember, x - h shifts the graph right, and x + h shifts it left.
• The parameter k controls the vertical translation (shift). +k shifts up, and -k shifts down.
• The parameter a controls the vertical stretch/compression and reflection.
  • If |a| > 1, the graph is stretched vertically (it becomes "narrower").
  • If 0 < |a| < 1, the graph is compressed vertically (it becomes "wider").
  • If a < 0, the graph is reflected across the x-axis (the "V" opens downwards).

The following resource provides a concise written summary of these transformations.

Putting It All Together: A Step-by-Step Example

Let's graph f(x) = -2|x - 1| + 4.

1. Find the Vertex (h, k):

   • Comparing our function to a|x - h| + k, we have a = -2, h = 1, and k = 4.
   • The vertex is at (1, 4).

2. Determine the Shape/Orientation:

   • Since a = -2 is negative, the graph opens downwards.
   • Since |a| = 2 is greater than 1, the graph is vertically stretched by a factor of 2. This means the slopes of the two lines forming the "V" will be +2 and -2.

3. Find the Intercepts:

   • y-intercept (set x = 0):
     y = -2|0 - 1| + 4 = -2|-1| + 4 = -2(1) + 4 = 2.
     The y-intercept is at (0, 2).
   • x-intercepts / Roots (set y = 0):
     0 = -2|x - 1| + 4
-4 = -2|x - 1|
2 = |x - 1|
     This gives two possibilities:
     x - 1 = 2 --> x = 3
x - 1 = -2 --> x = -1
     The x-intercepts are at (3, 0) and (-1, 0).

4. Sketch the Graph: Plot the vertex and intercepts, then draw the downward-opening V-shape.

The video below works through this exact example, visually demonstrating each transformation step by step.

2. Graphing Basic Square Root Functions

A square root function is one that contains a square root with the independent variable inside the root symbol (the radicand). The parent function is f(x) = √(x).

A crucial feature of the square root function is its domain. Since we cannot take the square root of a negative number in the real number system, the expression inside the square root must be greater than or equal to zero.

• For the parent function f(x) = √(x), the domain is x ≥ 0.
• The range (the set of possible y-values) is also y ≥ 0.

Let's watch a short introduction to the parent square root function and its key features.

Transformations of Square Root Functions

The transformation rules we know and love apply here as well. The general form is:
$f(x)=a\sqrt{x-h}+k$

• The point (h, k) is the starting point of the graph.
• h is the horizontal translation (shift).
• k is the vertical translation (shift).
• a controls the vertical stretch/compression and reflection across the x-axis.

The following resource from a textbook provides an excellent, structured summary of these transformations.

Step-by-Step Example with Domain & Range

Let's graph f(x) = \sqrt{x - 1} + 2.

1. Find the Starting Point (h, k):

   • Comparing our function to a√(x - h) + k, we have a = 1, h = 1, and k = 2.
   • The starting point is at (1, 2).

2. Find the Domain:

   • The expression inside the square root must be non-negative.
   • x - 1 ≥ 0
   • x ≥ 1. So, the domain is [1, ∞). This tells us the graph starts at x=1 and extends to the right.

3. Find the Range:

   • The starting y-value is k = 2. Since a=1 is positive, the graph goes upwards from there.
   • The range is [2, ∞).

4. Find Intercepts (if any):

   • y-intercept (set x = 0): y = √(0 - 1) + 2. Since x=0 is not in our domain (x ≥ 1), there is no y-intercept.
   • x-intercept (set y = 0): 0 = √(x - 1) + 2. This gives √(x - 1) = -2. Since the principal square root cannot be negative, there is no x-intercept.

5. Sketch the Graph: Start at the point (1, 2) and draw a curve with the same basic shape as the parent function, moving up and to the right. To get another point for accuracy, you can pick a convenient x-value in the domain, like x=5:
   y = √(5 - 1) + 2 = √4 + 2 = 2 + 2 = 4. So the point (5, 4) is on the graph.

The next video segment provides another clear, worked example, tying together the graph, domain, and range.

Conclusion

Today we've seen how the powerful concept of transformations can be applied to graph new families of functions. By identifying a parent function and its key features—like the vertex of an absolute value function or the starting point of a square root function—we can use our knowledge of shifts, stretches, and reflections to sketch any transformed version.

Key Takeaways:

• The graph of an absolute value function f(x) = a|x - h| + k has a V-shape with its vertex at (h, k). The parameter a determines the orientation (up/down) and stretch.
• The graph of a square root function f(x) = a\sqrt{x - h} + k is a curve with a starting point at (h, k).
• A critical feature of square root functions is the restricted domain, found by setting the expression inside the square root to be greater than or equal to zero.
• The process of identifying key features (vertex/starting point, intercepts) and using transformations is a versatile strategy for understanding and graphing many types of functions.

Preview of the Next Lesson:
In our next lesson, "Translate word problems into mathematical models using linear and quadratic functions," we will shift our focus to application. You'll learn how to take a real-world scenario, translate it into a linear or quadratic equation, and use the functions we've studied to solve the problem.