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Chapter 2: Problem 96

Graph each function. $$ f(x)=\left\\{\begin{aligned} x^{2} \text { for } x & \geq 0 \\ |x| \text { for } x &<0 \end{aligned}\right. $$

Short Answer

Expert verified
Graph two parts: parabolic for \( x \geq 0 \) and V-shape for \( x < 0 \).

Step by step solution

01

Understand the Piecewise Function

The function provided is piecewise, meaning it is defined differently depending on whether the value of \( x \) is greater than 0 or less than 0. For \( x \geq 0 \), the function is \( f(x) = x^2 \), which is a quadratic function. For \( x < 0 \), the function is \( f(x) = |x| \), which represents the absolute value function.
02

Graph the Quadratic Function for \( x \geq 0 \)

To graph \( f(x) = x^2 \) for \( x \geq 0 \), plot points starting from \( x = 0 \). The shape is a parabola opening upwards beginning at the origin (0,0). The key points are (0,0), (1,1), (2,4), and continue similarly for other positive \( x \) values.
03

Graph the Absolute Value Function for \( x < 0 \)

For \( x < 0 \), graph \( f(x) = |x| \). This function equals \(-x\) for negative \( x \) values, resulting in a V-shape starting from the origin. Key points to plot include (-1,1), (-2,2), (-3,3). This portion only covers the left side of the y-axis as \( x < 0 \).
04

Combine and Sketch the Graph

Combine the plots from Steps 2 and 3. The parabola starts at the origin and extends to the right for \( x \geq 0 \), and the V-shape from the absolute value function extends to the left for \( x < 0 \). Ensure there is a smooth transition at the origin, where the vertex of the parabola and the V meet.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadratic Functions
Quadratic functions are a fundamental concept in mathematics. They are expressed in the standard form of \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). In our piecewise function, the quadratic part is \( f(x) = x^2 \), which simplifies to a base quadratic with no linear or constant terms.

This specific quadratic function, where \( a = 1 \), is a simple parabola. It opens upwards because the leading coefficient \( a \) is positive. The graph of \( f(x) = x^2 \) for \( x \geq 0 \) is an upward-sloping curve starting from the origin (0,0) and extending infinitely to the right.

Quadratic functions exhibit certain key features:
  • They are always symmetrical around their vertex. For \( f(x) = x^2 \), the vertex is at the origin \((0,0)\).
  • The graph takes the shape of a U or parabola. For \( x^2 \), the parabola opens upward, indicating an increasing function as \( x \) moves away from zero towards positive values.
  • The rate at which the function value increases is exponential - doubling \( x \) leads to quadrupling the value of \( f(x) \).
Absolute Value Functions
Absolute value functions are another interesting type of function with unique graph shapes. The absolute value of \( x \), denoted as \( |x| \), equals \( x \) when \( x \geq 0 \) and \( -x \) otherwise. In our exercise, the absolute value function is utilized for \( x < 0 \), forming a key part of the piecewise function.

For \( x < 0 \), the graph of \( f(x) = |x| \) transforms into a linear function \( f(x) = -x \), which produces a line. This line starts at the origin and slopes downwards to the left, forming one side of a characteristic V-shape.

Key characteristics of absolute value functions include:
  • The graph forms a V-shape, with the vertex at the origin (0,0) for \( f(x) = |x| \).
  • The slope of the left side of the V is negative, which reflects the equation \( f(x) = -x \) used for \( x < 0 \).
  • This function is linear in each piece, unlike quadratics, meaning the rate of change is constant within each region.
Graphing Functions
Graphing functions involves displaying them visually on a coordinate plane to understand their behavior and interactions. When dealing with piecewise functions, such as the one in this exercise, you need to graph each part separately and then combine them.

To graph a piecewise function:
  • Identify the domain for each piece of the function. For our function, \( f(x) = x^2 \) applies when \( x \geq 0 \), and \( f(x) = |x| \) when \( x < 0 \).
  • Plot key points for each section. For \( x^2 \), plot (0,0), (1,1), and (2,4). For \( |x| \), plot (-1,1), (-2,2), and (-3,3).
  • Draw the individual curves and lines according to the determined points. The parabola and V-shape ensure a smooth transition at their meeting point at the origin.
Graphing these functions not only helps in visualizing mathematical equations but also in interpreting the interactions and continuity at transition points, such as at \( x = 0 \) where the segments meet. Adjust your plots carefully to maintain accurate slopes and curvatures as dictated by the equations.

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