Problem 19 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 13-26: Sketch, on the same coordinate plane, the graphs of \(f\) for the given values of \(c\). (Make use of symmetry, shifting, stretching, compressing, or reflecting.) $$ f(x)=\frac{1}{2} \sqrt{x-c} ; \quad c=-2,0,3 $$

Step-by-Step Solution

Answer

Each graph is a rightward-opening square root curve, shifted horizontally based on the value of \( c \).

Step 1: Identify the Base Function

The base function provided is \( f(x) = \frac{1}{2} \sqrt{x - c} \). The function depends on the parameter \( c \), which will shift the graph along the x-axis.

Step 2: Analyze the Effects of c

The value of \( c \) in \( f(x) = \frac{1}{2} \sqrt{x - c} \) dictates a horizontal shift. When \( c = -2 \), the function shifts 2 units to the left. For \( c = 0 \), it represents no horizontal shift. Finally, for \( c = 3 \), the graph shifts 3 units to the right.

Step 3: Graph for c = -2

Graph the function \( f(x) = \frac{1}{2} \sqrt{x + 2} \). Start at \( x = -2 \) because the square root requires non-negative inputs. The graph is a rightward-opening square root curve starting at \( x = -2 \).

Step 4: Graph for c = 0

Plot the function \( f(x) = \frac{1}{2} \sqrt{x} \). This standard square root graph begins at \( x = 0 \) and is stretched vertically by a factor of \( \frac{1}{2} \), producing a slightly flattened curve.

Step 5: Graph for c = 3

Sketch the graph for \( f(x) = \frac{1}{2} \sqrt{x - 3} \). This graph begins at \( x = 3 \) and follows the same stretched form as before. The graph opens to the right from \( x = 3 \).

Step 6: Combine Graphs on Same Plot

Overlay the functions from steps 3, 4, and 5 on the same coordinate plane. Each graph maintains its characteristic shape but starts from different x-values corresponding to \( c = -2, 0, \text{and } 3 \).

Key Concepts

Square Root Function

A square root function is a type of function that includes a square root operation in its expression. For example, the function \( f(x) = \sqrt{x} \) represents a basic square root graph. This graph is familiar to many; it starts at the origin (0,0) and increases without bound as \( x \) increases. The basic square root function has its domain starting from zero to positive infinity, as square roots are not typically defined for negative numbers over the real number set. Square root functions have a characteristic shape: a curve that begins at a point on the x-axis and gradually rises as it moves to the right, resembling a sideways parabola. For our specific function \( f(x) = \frac{1}{2} \sqrt{x-c} \), the square root part \( \sqrt{x-c} \) determines the general shape of the graph, while other modifications will transform its appearance.

Horizontal Shifts

Horizontal shifts are transformations that move a graph left or right along the x-axis. In the function \( f(x) = \sqrt{x-c} \), changing the value of \( c \) will shift the graph horizontally.

• If \( c \) is positive, the graph shifts to the right by \( c \) units.
• If \( c \) is negative, the graph moves to the left by the absolute value of \( c \) units.

In our exercise, this is demonstrated as follows:- \( c = -2 \): The function \( f(x) = \frac{1}{2} \sqrt{x + 2} \) highlights a shift 2 units to the left. - \( c = 0 \): Simply \( f(x) = \frac{1}{2} \sqrt{x} \), representing no horizontal shift.- \( c = 3 \): This shifts the graph 3 units to the right in \( f(x) = \frac{1}{2} \sqrt{x - 3} \).Understanding horizontal shifts helps in plotting the starting point of the graph precisely on the x-axis.

Graph Sketching

Graph sketching combines understanding of transformations with actual plotting. Sketching helps visualize how changes in the function equation reflect on the graph's shape and position.When sketching \( f(x) = \frac{1}{2} \sqrt{x-c} \):

• Start by finding where the square root expression equals zero; this is the starting point on the x-axis (\( x=c \)).
• Recognize that the half coefficient \( \frac{1}{2} \) stretches the graph vertically. Each point's height is half what it would be in the standard square root graph.
• Draw the characteristic sideways curve, ensuring it never descends but gradually rises to the right.

Always consider any transformations in the context of the base graph. For deep practice, sketch multiple graphs on the same coordinate system, noting any differences caused by transformations.

Function Transformations

Function transformations are methods to modify a graph's shape, size, and position. Common transformations affecting graphs include shifts, stretches, compressions, and reflections.

• Horizontal Shifts: Move the function left or right. As noted, our function's parameter \( c \) directly influences this shift.
• Vertical Stretches or Compressions: Achieved with coefficients outside the square root (e.g., \( \frac{1}{2} \) makes the graph rise more slowly).
• Reflections: These flip a graph across an axis, but are absent in our specific function example.

By understanding how each transformation affects the graph, you can predict and accurately sketch any variations of the original function. Mastery of these transformations supports deeper comprehension of more complex functions.