Problem 11 - Algebra 2

Question

For each function, identify the translation of the parent function. Then graph the function. \(y=|x+2|\)

Step-by-Step Solution

Answer

The function y = |x + 2| translates the parent function y = |x| two units to the left. Graph this by moving the vertex of the parent function's V-shape from (0,0) to (-2,0) and drawing the rays with the same slope from the new vertex.

Step 1 - Identify the Translation

To determine the translation of the parent function for the given function, we compare it with the parent function's form, which is of the shape |x|. In the given function, y = |x + 2|, the +2 inside the absolute value translates the graph horizontally. Since it is +2, this means the graph of the parent function y = |x| shifts 2 units to the left.

Step 2 - Draw the Parent Function

Start by sketching the parent function y = |x|, which consists of two rays. The first ray starts at the origin (0,0) and goes up with a slope of 1 to the right, and the second starts at the origin and goes up with a slope of -1 to the left.

Step 3 - Apply the Horizontal Translation

To graph y = |x + 2|, shift the graph of the parent function, y = |x|, 2 units to the left. This means the vertex of the graph, which was originally at the origin, will now be at (-2,0). Draw the two rays starting from the point (-2,0), with the same slopes as the parent function.

Step 4 - Verify the Translated Graph

Check that the graph correctly mirrors over the new vertex at (-2,0) and that the shape remains the same as the parent function, except for the shift to the left.

Key Concepts

Parent Function

Understanding the parent function is crucial when learning about transformations in algebra. The parent function for absolute value functions is represented as
\( y = |x| \).

It consists of two linear parts that form a V shape. The point of intersection of these two lines, known as the vertex, is located at the origin (0,0). This function is even, meaning it’s symmetric about the y-axis. The slopes of the two rays that form the V are 1 and -1, respectively. These basic traits of the parent function are essential as they remain unchanged during transformations; only their position and orientation might change.

Horizontal Translation

Horizontal translation shifts the graph of the function to the left or right on the coordinate plane without altering its shape or orientation.

Understanding Horizontal Shifts

To perform a horizontal translation of the absolute value function, you adjust the input variable x. When you see a function like \( y = |x + k| \), where \( k \) is a positive or negative number, this indicates a horizontal shift. A positive value of \( k \) shifts the graph to the left by \( k \) units, whereas a negative value shifts it to the right by \( |k| \) units. It’s important to remember that the inside addition or subtraction affects the x-coordinate of the vertex of the graph.

Absolute Value Transformations

Transformations of the absolute value functions involve shifting, reflecting, stretching, or compressing the parent function, while maintaining the overall V shape of the graph. These transformations can be identified by modifications to the standard form of the absolute value equation.

Types of Transformations

The general form of an absolute value function is \( y = a|bx + c| + d \), where 'a' affects vertical stretching or compressing and may reflect the graph across the x-axis, 'b' affects the slope of the lines and may reflect the graph across the y-axis, ‘c’ deals with horizontal translations, and 'd' deals with vertical translations. Recognizing these parts of the equation allows you to predict graph behavior before even plotting it on the coordinate plane.

Vertex of a Graph

The vertex of an absolute value graph is a critical point that defines the sharpest turn or the point where the graph changes direction. In the parent function \( y = |x| \), the vertex is at the origin.

Role of the Vertex in Translations

When absolute value functions are translated horizontally or vertically, the vertex moves accordingly. For the function \( y = |x + c| \), a positive 'c' moves the vertex 'c' units to the left, while a negative 'c' moves it 'c' units to the right. Similarly, for the function \( y = |x| + d \), a positive 'd' value lifts the vertex 'd' units up, and a negative 'd' lowers it 'd' units down. The position of the vertex is also an indicator of where to start graphing the function as it’s a fixed point from which the V-shape emanates.