Problem 10 - Precalculus with Limits

Question

For each function, sketch (on the same set of coordinate axes) a graph of each function for \(c = -3\), \(-1\), \(1\), and \(3\). (a) \( f(x) = \left\\{ \begin{array}{ll} x^2 + c, & \mbox{ \) x < 0 \(} \\ -x^2+c, & \mbox{ \) x \geq 0 \(} \end{array} \right.\) (b) \( f(x) = \left\\{ \begin{array}{ll} (x + c)^2, & \mbox{ \) x < 0 \(} \\ -(x + c)^2, & \mbox{ \) x \geq 0 \(} \end{array} \right.\)

Step-by-Step Solution

Answer

The sketching of a piecewise defined function requires care to account for every part of the definition domain and parameter being used. In both (a) and (b) function nodes, the graphs are parabolas with direction and position shifts depending on 'c'. In every case, whether opening upwards or downwards, the functions accurately represent the given domains and 'c' values.

Step 1: Interpret the Piecewise Functions

Notice that every given function in this exercise is a piecewise function. A piecewise function is a function that is defined by two or more rules or pieces, each applying to a different part of the domain. Here, each function behaves differently to the left and right of x = 0, with 'c' acting as a vertical shift.

Step 2: Graph the Function for (a)

To graph \(f(x) = \left\{ \begin{array}{ll} x^2 + c, & \mbox{ for } x < 0 \ -x^2+c, & \mbox{ for } x \geq 0 \end{array} \right.\),start by treating each piece of the function separately. For each given value of 'c', when x is less than zero, the graph is a parabola opening upwards shifted 'c' units. And when x is greater than or equal to zero, the graph is a parabola opening downwards, also shifted 'c' units. Sketch this for c = -3, -1, 1, and 3.

Step 3: Graph the Function for (b)

For \(f(x) = \left\{ \begin{array}{ll} (x + c)^2, & \mbox{ for } x < 0 \ -(x + c)^2, & \mbox{ for } x \geq 0 \end{array} \right.\),again, examine each piece of the function individually. For each value of 'c', when x is less than zero, the graph is an upwards opening parabola shifted 'c' units to the left. And when x is greater than or equal to zero, it becomes a downwards opening parabola, shifted 'c' units to the left. Sketch these graphs for each given 'c' value.

Key Concepts

Vertical Shift

A vertical shift is a transformation that moves a graph up or down without altering its shape. Imagine holding the graph in your hands and simply sliding it up or down the coordinate plane. In the context of piecewise functions, the constant 'c' in this exercise represents this vertical shift. For each segment of the function, adding c shifts the graph upwards if c is positive and downwards if c is negative.
For example, in the function part for \(x^2 + c\), if c equals 3, every point on the parabola shifts 3 units up. Conversely, if c is -3, the parabola shifts 3 units down. This transformation helps analyze how differently positioned functions behave under the same x-interval conditions.

Parabola Graphs

Parabolas are u-shaped graphs typical of quadratic functions. They have a vertex, which is the lowest or highest point depending on whether they open upwards or downwards. In this exercise, the functions \(x^2 + c\) and \(-x^2 + c\) describe parabolas.
When \(x < 0\), we use \(x^2 + c\) for an upwards opening parabola, and for \(x \geq 0\), \(-x^2 + c\) gives a downwards parabola. Meanwhile, transformations in part (b) involve horizontally shifting the function before squaring it, impacting where the parabola appears along the x-axis but maintaining its u-shape. Each choice of c shifts the vertex of the parabola vertically as described.

Domain and Range

The domain of a function is all possible input values (x-values) it can take, while the range is all possible output values (y-values). For these piecewise functions, the domain is broken into two parts, depending on the value of x.

• For \(x < 0\), the domain covers all negative x-values.
• For \(x \geq 0\), it includes zero and all positive x-values.

The range changes with each vertical shift c. Generally, when \(f(x) = x^2 + c\), the range will be \(y \,\geq \,c\), and for \(f(x) = -x^2 + c\), it is \(y \,\leq \,c\). Knowing the domain and range is key for sketching functions accurately.

Function Sketching

Function sketching involves creating a visual representation of a function based on its equation. Begin by identifying key features of the function like the vertex of a parabola, the direction it opens, and any shifts involved.
For piecewise functions, it's crucial to graph each piece separately, respecting their particular domain restrictions. This requires identifying breakpoints like x = 0 in this exercise, where the function rule changes.

• When \(c = -3\), plot the graph of each piece adjusted down by 3 units.
• With \(c = 3\), raise each piece 3 units higher.

Through consistent practice, you can hone your skills to accurately visualize and plot even complex function types.