Problem 26 - Algebra and Trigonometry

Question

23–26 ? Explain how the graph of g is obtained from the graph of f. (a) \(f(x)=|x|, \quad g(x)=3|x|+1\) (b) \(f(x)=|x|, \quad g(x)=-|x+1|\)

Step-by-Step Solution

Answer

(a) Stretch vertically by 3 and shift up by 1. (b) Shift left by 1 and reflect across x-axis.

Step 1: Understand the Basic Graph

The function \( f(x) = |x| \) represents the graph of an absolute value function, which is a V-shaped graph centered at the origin (0,0) and opens upwards.

Step 2: Apply Vertical Stretch and Translation

For part (a), the function \( g(x) = 3|x| + 1 \) means we apply a vertical stretch by a factor of 3 to \( f(x) \) and then translate it upward by 1 unit. This stretches the graph, making it taller and shifts it upwards.

Step 3: Perform a Horizontal Shift and Reflect

For part (b), the function \( g(x) = -|x+1| \) involves translating \( f(x) \) horizontally to the left by 1 unit (since \( |x+1| \) shifts the graph to the left), and then reflecting it across the x-axis due to the negative sign in front.

Key Concepts

Absolute Value Function

An absolute value function is represented by the expression \(f(x) = |x|\). This function is easy to recognize with its unique V-shaped graph. The graph always consists of two linear rays that meet at a point called the vertex. For the basic form of the absolute value function, this vertex is located at the origin (0,0). The graph opens upwards and symmetrically extends on both sides of the y-axis.

• Each side of the V shape is a straight line.
• The vertex is the closest point of the graph to the x-axis.
• The graph is not one-to-one, as it reflects over the y-axis.

Knowing this basic structure helps in understanding how transformations change the original graph.

Vertical Stretch

A vertical stretch occurs when the graph of a function expands or contracts along the y-axis. This transformation affects the steepness of the graph. For example, for part (a) where \(g(x) = 3|x| + 1\), the number 3 in front of the absolute value function indicates a vertical stretch by a factor of 3.

• The graph becomes steeper.
• Each point on the graph is moved 3 times further from the x-axis.
• This does not affect the x-coordinates of the points, only their y-coordinates.

This transformation makes the graph appear taller, while the overall shape and direction remain unchanged.

Translation

Translation is a type of transformation that shifts the entire graph either vertically, horizontally, or both, without altering its shape or orientation. In part (a) of the problem, \(g(x) = 3|x| + 1\) experiences two transformations: a vertical stretch and a translation. The \(+1\) indicates an upward shift by 1 unit.

• This moves every point on the graph vertically by the same amount.
• The shifted function maintains its original V shape.
• The vertex moves from (0,0) to (0,1).

Translations can make it easier to place the graph in a desired position on the coordinate plane.

Reflection

Reflection is a transformation resulting in a mirror image of the graph, typically over the x-axis or y-axis. For part (b), \(g(x) = -|x+1|\), the negative sign in front of the absolute value function indicates a reflection over the x-axis.

• The V shape opens downwards instead of upwards.
• Every point's y-coordinate is multiplied by -1, flipping it vertically.
• Despite this flip, the absolute value of x remains unaffected.

This transformation can dramatically change the visual interpretation and behavior of the graph.

Horizontal Shift

Horizontal shifts move the graph left or right along the x-axis. This shift is determined by the expression inside the absolute value. In \(g(x) = -|x+1|\) of part (b), the \(+1\) inside \(|x+1|\) indicates a shift left by 1 unit.

• The graph moves towards the negative side of the x-axis.
• The vertex shifts from (0,0) to (-1,0).
• Only the x-coordinates are affected, while the graph's symmetry and steepness remain unchanged.

Understanding horizontal shifts enables precise placement of graphs within the coordinate plane.