Problem 24 - Calculus with Concepts in Calculus

Question

Sketch the graph of the function. $$ f(x)=\left\\{\begin{array}{l} x^{2} \text { for } x<0 \\ -x \text { for } x \geq 0 \end{array}\right. $$

Step-by-Step Solution

Answer

The graph consists of a parabola for \(x<0\) and a line with slope -1 for \(x\geq0\).

Step 1: Understand the Function Definition

The function \(f(x)\) is a piecewise function with two different expressions based on the value of \(x\). For \(x < 0\), \(f(x) = x^2\), which is a parabola opening upwards. For \(x \geq 0\), \(f(x) = -x\), which is a line with a negative slope.

Step 2: Analyze the Function for \(x < 0\)

For values of \(x < 0\), \(f(x) = x^2\). The graph will be a parabola that opens upwards. All points will lie in the second quadrant because \(x\) is negative, but \(x^2\) is positive, creating the shape of half a parabola.

Step 3: Analyze the Function for \(x \geq 0\)

For values of \(x \geq 0\), \(f(x) = -x\). The graph will be a straight line with a slope of -1, starting from the origin. This line will decrease through the first quadrant and continue into the fourth quadrant.

Step 4: Determine Key Points and Intersections

1. Intersection at the origin: The parabola does not include \(x=0\) but ends at it with a value of \(f(0^-)=0\). The line includes the point \((0,0)\) as \(f(0)=0\).2. The slope of -1 for \(x \geq 0\) confirms a straight line through points like \((0,0), (1,-1), (2,-2)\).

Step 5: Sketch the Graph

Draw the graph based on the analysis:1. Plot half a parabola that stops at the origin, covering \(x < 0\).2. Begin a line at the origin with a slope of -1, extending through the first quadrant.3. The graph consists of a smooth connected parabola on the left and a downward slope line on the right.

Key Concepts

Graphing Functions

Graphing functions helps us visualize how a function behaves across different values of \(x\). A function's graph can take many forms, such as lines, curves, or combinations of both, depending on the function's definition. In the case of the given piecewise function \(f(x)\), graphing it requires understanding its distinct parts as separate entities before combining them.

When graphing piecewise functions, remember to:

• Identify the range of \(x\) for each piece of the function.
• Understand the shape and behavior of each part, as determined by its function type (e.g., linear, quadratic).
• Mark key points and check for any intersections or transitions between the segments.

In our example, each piece of the function \(f(x)\) is defined by a specific rule, making it straightforward to plot each separately and then piece them together on a single graph. Doing so offers a deeper understanding of the unified behavior of these segments in relation to each other.

Parabolas

Parabolas are an integral part of quadratic functions. The segment of the piecewise function \(f(x) = x^2\) for \(x < 0\) is a perfect example. Parabolas have the characteristic "U" shape, and for \(x^2\), the parabola opens upwards.

For the function \(f(x) = x^2\) when \(x < 0\):

• The vertex of the parabola is the origin (0,0), but it only includes points to the left of this vertex because \(x\) is negative.
• The axis of symmetry is the y-axis, a line that divides the parabola into two mirror images.
• All \(y\) values will be positive, forming a half-parabola in the second quadrant of the graph.

Parabolas add flexibility to our graphs, allowing for smooth transitions and curvature. When dealing with a piecewise function, it's essential to accurately plot each parabola section while respecting its domain limitation, as seen in this example.

Linear Functions

Linear functions create straight lines on a graph, characterized by their constant rate of change or slope. In the piecewise function, the section \(f(x) = -x\) for \(x \geq 0\) is linear with a slope of -1.

Here's what to note about this linear part:

• The line starts at the origin (0,0) since \(f(0) = 0\).
• It descends diagonally through the first and fourth quadrants due to its negative slope.
• Each step along the line involves a decrease in \(y\) for each increase in \(x\).

Linear functions are predictable and straightforward. They provide a contrast to the more complex shapes like parabolas, as seen in this graph. Understanding how the linear section interacts and connects with other parts of a piecewise function is crucial for an accurate, comprehensive graphing effort.