Problem 4 - College Algebra

Question

How does the graph of the absolute value function compare to the graph of the quadratic function, \(y=x^{2},\) in terms of increasing and decreasing intervals?

Step-by-Step Solution

Answer

Both functions decrease for \( x < 0 \) and increase for \( x > 0 \), but \( y = |x| \) forms a V-shape, while \( y = x^2 \) forms a U-shape.

Step 1: Graph the Absolute Value Function

The graph of the absolute value function is written as \( y = |x| \). This function creates a V-shape, with the vertex located at the origin (0,0). The graph consists of two linear segments: one with a positive slope to the right of the vertex and one with a negative slope to the left.

Step 2: Graph the Quadratic Function

The quadratic function \( y = x^2 \) is a parabola with its vertex also at the origin (0,0). This graph is symmetric about the y-axis and is U-shaped. Both arms of the parabola go upwards as \( x \) moves away from the vertex in either direction.

Step 3: Identify the Increasing and Decreasing Intervals for the Absolute Value Function

For \( y = |x| \), the function decreases when \( x < 0 \) as the slope is negative, and it increases when \( x > 0 \) as the slope is positive. At \( x = 0 \), the rate of change switches from negative to positive.

Step 4: Identify the Increasing and Decreasing Intervals for the Quadratic Function

The function \( y = x^2 \) decreases when \( x < 0 \), as the left side of the parabola moves downwards toward the vertex, and increases when \( x > 0 \), as the right side moves upwards away from the vertex. The rate of change is zero at \( x = 0 \), where the function reaches its minimum.

Step 5: Compare the Increase and Decrease Intervals of Both Functions

Both functions decrease when \( x < 0 \) and increase when \( x > 0 \). However, \( y = |x| \) changes from decreasing to increasing in a sharp V-shape, while \( y = x^2 \) does so smoothly in a U-shape.

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial function where the highest degree of the variable is two. It has the general form: \[ y = ax^2 + bx + c \] where \( a \), \( b \), and \( c \) are constants with \( a eq 0 \). The graph of a quadratic function is a parabola. Some important features of parabolas include:

• Vertex: The highest or lowest point on the graph, which for the quadratic \( y = x^2 \), is at the origin (0,0).
• Axis of Symmetry: A line that vertically passes through the vertex, often represented as \( x = h \) where \( h \) is the x-coordinate of the vertex.
• Direction: If \( a > 0 \), the parabola opens upwards; if \( a < 0 \), it opens downwards.

Understanding these basic features is essential to graphing and analyzing quadratic functions.

Graphing Functions

Graphing functions is an essential skill in mathematics that allows us to visualize the behavior and characteristics of different types of functions. For absolute value functions and quadratic functions:

• The graph of the absolute value function \( y = |x| \) appears as a V-shape, which is symmetric regarding the y-axis.
• The quadratic function \( y = x^2 \) creates a U-shaped parabola centered at the origin.
• Both graphs are useful for understanding concepts such as symmetry, vertices, and the impact of transformations on graph shapes.

Graphing these functions carefully helps identify essential details like intercepts and rates of change, providing deeper insights into their behavior over different intervals.

Increasing and Decreasing Intervals

Determining where a function increases or decreases helps in understanding its overall behavior. For the absolute value function \( y = |x| \):

• The function is decreasing when \( x < 0 \) as the slope of the line is negative.
• Once \( x \) reaches zero, it stops decreasing, and as \( x > 0 \), the function starts increasing since the slope switches to positive.

For the quadratic function \( y = x^2 \):

• It decreases when \( x < 0 \), as the path of the parabola moves downward towards its vertex.
• At \( x = 0 \), the function reaches its lowest point, known as the minimum.
• For \( x > 0 \), the function begins to increase, moving upwards away from the vertex.

Comparing both, both functions follow similar patterns in terms of increasing and decreasing intervals, providing insights into their smooth versus sharp transitional behaviors.

Symmetry in Graphs

Symmetry in graphs simplifies the process of understanding and predicting how a graph behaves over its domain. The absolute value function \( y = |x| \) and the quadratic function \( y = x^2 \) both display symmetry:

• The absolute value function is symmetric about the y-axis, creating a mirror image on both sides, resulting in its characteristic V-shape.
• The quadratic function also exhibits symmetry about the y-axis. This symmetry creates the U-shape that opens at the vertex and both sides are congruently shaped.

This symmetry is due to how each function responds to positive and negative inputs, remaining unaffected by changes in the sign of \( x \). Such symmetry allows for easier calculation and verification when analyzing and comparing different types of functions.