Question

Sketch the graph of the function. Choose a scale that allows all relative extrema and points of inflection to be identified on the graph. \(y=\left\\{\begin{array}{r}x^{2}+4, x<0 \\ 4-x, x \geq 0\end{array}\right.\)

Short answer

The graph of the function \(y=\left\{\begin{array}{r}x^{2}+4, x<0 \\ 4-x, x \geq 0\end{array}\right.\) is made up of a quarter parabola for \(x<0\) and a straight line decreasing to negative infinity for \(x \geq 0\). The two parts of the graph meet at \((0, 4)\). There are no points of discontinuity, extrema, or inflection points.

Step-by-step solution

Analyzing and sketching the first part of the piecewise function

The first part of the piecewise function is \(x^{2}+4\), valid for \(x<0\). This is a simple parabola shape quadratic function, whose minimum point occurs at \(x=0\) and \(y=4\). Because the domain of this function is \(x<0\), we only consider the left side of the parabola where \(x<0\). Sketch this part of the graph.

Analyzing and sketching the second part of the piecewise function

The second part of the piecewise function is \(4-x\), where \(x \geq 0\). This is a linear function with a negative slope, which means it is a decreasing straight line. Its value at \(x=0\) is \(y=4\), where it starts and goes downwards to the right. Sketch this on the graph, starting at the point \((0, 4)\).

Combine the sketches

Now, combine the two parts of the sketches. Make sure that they meet at \(x=0\), \(y=4\). For \(x<0\), the graph is a quarter parabola above the x-axis and for \(x \geq 0\) it is a straight line descending downwards.

Key concepts

Graph Sketching

Graph sketching is an essential skill in mathematics and involves drawing the approximate shape of a graph based on its mathematical characteristics. When working with piecewise functions, it's crucial to understand the behavior of each segment to ensure an accurate sketch.

To sketch a function accurately:

• Identify the type of function in each defined interval.
• Determine key features like intercepts, asymptotes, and vertexes in those intervals.
• Analyze how each piece connects or transitions into the next.
• Choose a scale that allows you to plot important features like relative extrema and inflection points.

For example, the function given in the exercise is split into two parts: one is a quadratic function, and the other is a linear function. Knowing these types helps predict their sketch on the graph efficiently. The first part, a parabola, is curved, while the second part appears as a straight line. At the boundary point where the function pieces transition, it's crucial to ensure they meet accurately, as this affects the graph's accuracy.

Quadratic Functions

Quadratic functions are polynomial functions characterized by an equation of the form \(y = ax^2 + bx + c\). They graph as parabolas, which can open upwards or downwards depending on the sign of the coefficient \(a\).

Key features of a parabola include:

• The vertex, which is the maximum or minimum point of the curve.
• The axis of symmetry, a vertical line passing through the vertex.
• The direction of opening, determined by the sign of \(a\).

In this exercise, the part of the piecewise function \(x^2 + 4\) describes a parabola that has a minimum at \(x = 0, y = 4\). Since \(x < 0\) is the domain for this part, you only sketch the left-side of the parabola. It's essential to properly scale the y-axis so the vertex is clearly visible, aiding in understanding the graph's behavior.

Linear Functions

Linear functions are the simplest kind of functions and are represented by equations of the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. The graph of a linear function is a straight line.

Important features of linear functions include:

• The slope, which indicates the steepness and direction of the line. A positive slope means the line ascends (increases), while a negative slope means it descends (decreases).
• The y-intercept, which is the point where the line crosses the y-axis.

For the exercise, the function \(4 - x\) is part of the piecewise function operating over \(x \geq 0\). Here, the slope \(-1\) indicates a downward trend. The line begins at \((0, 4)\) and descends to the right. This part of the graph complements the parabolic section by meeting at the point \((0, 4)\), ensuring a continuous graph across the two functions.