Graphs of absolute value functions

When studying "Shifting Absolute Value Graphs," focus on the following key points:

1. Understanding the Base Function: The basic absolute value function is $y = |x|$, which has a V-shape that opens upwards, with the vertex at the origin (0,0).

2. Vertical Shifts: Adding or subtracting a constant $k$ to the function results in a vertical shift:

   • $y = |x| + k$: Shifts the graph up by $k$.
   • $y = |x| - k$: Shifts the graph down by $k$.

3. Horizontal Shifts: Adding or subtracting a constant $h$ inside the absolute value expression shifts the graph horizontally:

   • $y = |x - h|$: Shifts the graph to the right by $h$.
   • $y = |x + h|$: Shifts the graph to the left by $h$.

4. Reflection: A negative sign in front of the absolute value function reflects the graph over the x-axis:

   • $y = -|x|$ flips the graph upside down.

5. Combining Shifts: You can combine horizontal and vertical shifts in the same function. For example:

   • $y = -|x - h| + k$: This reflects the graph, shifts it right by $h$, and then shifts it up by $k$.

6. Graphing Practice: Practice graphing various transformations of the absolute value function to understand how shifts change the graph's position and shape.

7. Identifying Key Features: Recognize how to identify the vertex after shifts and how it affects the domain and range of the function.

This foundational knowledge will help understand and manipulate absolute value functions effectively.

When studying "Scaling & Reflecting Absolute Value Functions," focus on these key points:

1. Basic Form: The standard form of an absolute value function is $f(x) = a|x - h| + k$, where:

   • $a$ determines the vertical stretch/compression and reflection.
   • $h$ shifts the function horizontally.
   • $k$ shifts the function vertically.

2. Vertical Scaling:

   • If $|a| > 1$, the function stretches vertically.
   • If $0 < |a| < 1$, the function compresses vertically.
   • If $a < 0$, the function reflects over the x-axis.

3. Horizontal Shifting:

   • The value of $h$ shifts the graph left (if $h > 0$) or right (if $h < 0$).

4. Vertical Shifting:

   • The value of $k$ shifts the graph up (if $k > 0$) or down (if $k < 0$).

5. Graphing Technique:

   • Start with the basic graph of $y = |x|$.
   • Apply transformations based on the values of $a$, $h$, and $k$.

6. Identifying Vertex:

   • The vertex of the function is at the point $(h, k)$.

7. Effects of Parameters:

   • Understand how each parameter affects the shape and position of the graph, allowing for predicting the graph of the function based on modifications made.

8. Example Problems:

   • Practice with various examples to solidify understanding of transformations and behaviors of different absolute value functions.

By mastering these points, you will have a comprehensive understanding of how scaling and reflection affect absolute value functions.

When studying "Scaling & Reflecting Absolute Value Functions: Graph," focus on the following key points:

1. Basic Shape: The parent absolute value function, f(x) = |x|, forms a V-shape with the vertex at the origin (0,0).

2. Scaling:

   • Vertical Scaling: The function f(x) = a|x| alters the vertical stretch or compression.
     • If |a| > 1, the graph stretches vertically.
     • If 0 < |a| < 1, it compresses vertically.
   • Horizontal Scaling: The function f(x) = |bx| affects horizontal stretch and compression.
     • If |b| > 1, the graph compresses horizontally.
     • If 0 < |b| < 1, it stretches horizontally.

3. Reflection:

   • Reflecting over the x-axis: If a is negative (f(x) = -|x|), the graph flips upside down.
   • Reflecting over the y-axis: f(x) = |−x| produces a graph identical to the parent function, as absolute value is even.

4. Transformations:

   • Vertical Shifts: f(x) = |x| + k moves the graph up (k > 0) or down (k < 0).
   • Horizontal Shifts: f(x) = |x - h| shifts the graph right (h > 0) or left (h < 0).

5. Vertex: The vertex of transformations can be found at the point (h, k) in the function f(x) = a|bx - h| + k.

6. Graphing Techniques: To accurately graph transformed functions, plot the vertex, apply scaling, and shifts, and then ensure symmetry about the vertical line through the vertex.

By mastering these concepts, you’ll be able to understand and graph various transformations of absolute value functions effectively.

When studying graphing absolute value functions, focus on the following key points:

1. Basic Form: The general form of an absolute value function is $f(x) = a|bx + c| + d$, where:

   • $a$ affects the vertical stretch/compression and reflection.
   • $b$ affects the horizontal stretch/compression.
   • $c$ determines the horizontal shift.
   • $d$ determines the vertical shift.

2. Vertex: The vertex of the graph is the point where the function changes direction, usually located at $(-\frac{c}{b}, d)$.

3. Axis of Symmetry: The graph is symmetrical about the vertical line $x = -\frac{c}{b}$.

4. Y-Intercept: To find the y-intercept, substitute $x = 0$ into the function.

5. Key Points: Plot the vertex and a few points on either side to form a "V" shape. The points can be found by choosing x-values around the vertex.

6. Behavior: The graph extends infinitely in both the upward direction for $a > 0$ and downward for $a < 0$.

7. Transformations: Understand how translations, reflections, and stretches/compressions affect the graph by modifying the parameters $a$, $b$, $c$, and $d$.

By mastering these concepts, you'll be able to effectively graph absolute value functions.