Problem 14 - Precalculus

Question

In Exercises \(13-20\), sketch the graph of the given piecewise-defined function. $$ f(x)=\left\\{\begin{array}{lll} x^{2} & \text { if } & x \leq 0 \\ 2 x & \text { if } & x>0 \end{array}\right. $$

Step-by-Step Solution

Answer

The graph transitions smoothly from a leftward parabola \(x^2\) at \(x \leq 0\) to a rightward line \(2x\) at \(x > 0\).

Step 1: Identify the Function Pieces

The given function is a piecewise function with two parts: \(x^2\) for \(x \leq 0\), and \(2x\) for \(x > 0\). Both parts need to be analyzed separately to understand their behaviors on the respective domains.

Step 2: Analyze the First Piece

For \(x \leq 0\), the function is \(f(x) = x^2\). This is a part of a parabola that opens upward. By considering only the left side (\(x \leq 0\)), it will look like a curve starting from the origin on the left, getting higher as \(x\) approaches negative infinity.

Step 3: Analyze the Second Piece

For \(x > 0\), the function is \(f(x) = 2x\). This is a linear function with a slope of 2. The line will start from just after \(x = 0\) moving rightward with increasing slope.

Step 4: Graph the Function Pieces

Now, sketch the graph based on the analyses. For \(x \leq 0\), draw a parabola opening upwards with the vertex at the origin. For \(x > 0\), sketch a line starting just beyond (0, 0) with a positive slope of 2.

Step 5: Check and Combine

At \(x = 0\), check transition: \(f(0) = 0^2 = 0\) which matches the start of the line \(2x\). As the conditions \(x \leq 0\) and \(x > 0\) are exclusive, there's no overlap where one function stops and the other begins. The graph will have a smooth transition at \(x = 0\).

Key Concepts

Graphing Functions

Graphing piecewise functions involves plotting each segment on the same graph while respecting their respective domains. This process gives a single, coherent visual representation that combines various types of functions.
To begin graphing, identify where each function applies on the x-axis. For example, with the given function, the quadratic segment is for values where \(x \leq 0\), and the linear segment is for \(x > 0\).
The trick is to clearly mark transition points. At \(x = 0\) in this function, the graph transitions from the parabola to the linear function.
When graphing each part:

• Ensure continuity by checking that each piece connects correctly on the graph.
• Draw solid lines for included boundaries and open circles for excluded points (though here, \(x = 0\) is included in the quadratic function and also serves as a start for the line).

This approach helps in properly visualizing and understanding the overall behavior of piecewise-defined functions.

Function Analysis

Analyzing piecewise functions means studying each component separately. This detailed look involves understanding their form, calculating values at critical points, and interpreting their behavior.
Start by analyzing individual formulas within their specified ranges. For our function, examine \(f(x) = x^2\) on \(x \leq 0\) which indicates a section of a parabola. This function has no linear component but a squared term dominating, causing the curve.

• Explore symmetry for parabolic sections around the vertex.
• Identify key points such as where \(x\) values equal zero.

Next, look at the linear part \(f(x) = 2x\) when \(x > 0\). Track steepness by the slope, calculated here as \(m = 2\).

• Pinpoint at which these calculations lead to significant change points, like intersection at\( (0,0) \).
• Merge insights from each function to anticipate the graph's shape and behavior.

Having this analysis helps in thoroughly understanding combined effects in piecewise functions.

Parabola

A parabola is a curve that looks like a U or an upside-down U. In mathematical terms, it's defined by quadratic equations such as \(f(x) = x^2\). In our piecewise function, this segment for \(x \leq 0\) replicates the left half.
This implies:

• The graph opens upwards when the coefficient of \(x^2\) is positive.
• It has a vertex, the starting point of the parabola, in this case at the origin \((0,0)\).

It is important to note that since it is part of a piecewise function, we only consider the behavior of the parabola as \(x\) gets more negative while maintaining its upward curve.
This part smoothly connects to any other function component at the boundary, like at \(x = 0\) for continuity.

Linear Functions

Linear functions create straight lines when graphed. They are defined by the formula \(f(x) = mx + b\), where \(m\) represents the slope, and \(b\) is the y-intercept.
Within the given piecewise function, the section \(f(x) = 2x\) for \(x > 0\) represents a line.

• The slope \(m = 2\) defines how steep the line is, indicating that for every unit increase in \(x\), \(f(x)\) rises by 2.
• The line doesn’t have a y-intercept here since it starts at \(x = 0\) and transitions immediately from the parabola.

To visualize, this line's trajectory continues upward to the right. Interruptions only occur at specific domain changes, maintaining linearity for the specified \(x > 0\) domain.
Understanding its construction helps accurately plot and predict its behavior alongside other function types within piecewise divisions.