Problem 8 - 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) \(f(x) = \sqrt{x} + c\) (b) \(f(x) = \sqrt{x - c}\) (c) \(f(x) = \sqrt{x - 3} + c\)

Step-by-Step Solution

Answer

The sketches of these functions show how the constant \(c\) affects the graph's position. For function (a) \(f(x) = \sqrt{x} + c\), \(c\) creates a vertical shift. For function (b) \(f(x) = \sqrt{x - c}\), \(c\) creates a horizontal shift, and for function (c) \(f(x) = \sqrt{x - 3} + c\), the function is shifted horizontally by 3 units to the right and vertically dependent on the value of \(c\).

Step 1: Graph function (a) \(f(x) = \sqrt{x} + c\) for different values of c

To graph this function, start with the basic square root function \(\sqrt{x}\). Then, for each constant value of \(c\), translate the graph vertically \(c\) units: up if \(c\) is positive, down if \(c\) is negative. For \(c = -3\), the graph is shifted down 3 units, for \(c = -1\), down 1 unit, for \(c = 1\), up 1 unit, and for \(c = 3\), up 3 units.

Step 2: Graph function (b) \(f(x) = \sqrt{x - c}\) for different values of c

For this function, the constant \(c\) affects the graph horizontally rather than vertically. Start with the basic square root function \(\sqrt{x}\). Then, for each constant value of \(c\), translate the graph horizontally \(c\) units to the right if \(c\) is positive, or left if \(c\) is negative. For \(c = -3\), the graph is shifted left 3 units, for \(c = -1\), left 1 unit, for \(c = 1\), right 1 unit, and for \(c = 3\), right 3 units.

Step 3: Graph function (c) \(f(x) = \sqrt{x - 3} + c\) for different values of c

This function shifts the graph both vertically and horizontally. Start with the basic square root function \(\sqrt{x}\), then shift it 3 units to the right (because of the \(-3\) in the square root) and \(c\) units up or down. For \(c = -3\), the graph is shifted 3 units to the right and down 3 units, for \(c = -1\), 3 units to the right and down 1 unit, for \(c = 1\), 3 units to the right and up 1 unit, and for \(c = 3\), 3 units to the right and up 3 units.

Key Concepts

Square Root Function

The square root function is one of the fundamental functions in mathematics and is often represented as \(f(x) = \sqrt{x}\). This particular function forms the basis of various transformations, especially those relating to graph translations. Its graph is characterized by a half-parabola that starts at the origin (point (0, 0)) and extends infinitely to the right with a gentle curve.

Understanding

• For positive values of \(x\), \(f(x) = \sqrt{x}\) gives the non-negative principal square root.
• The domain of the square root function is \(x \geq 0\) as square roots of negative numbers are not real.
• The range is all non-negative numbers \(y \geq 0\).

These properties make it a useful function when analyzing or sketching graphs that are subject to various transformations.

Vertical Shifts

Vertical shifts involve moving the graph of a function up or down along the y-axis. This transformation is represented in the equation by adding or subtracting a constant to the original function. Consider the function \(f(x) = \sqrt{x} + c\). Here, the constant \(c\) determines the direction and magnitude of the shift.

How It Works:

• If \(c\) is positive, the graph moves upward, which means that each point on the curve is moved \(c\) units higher.
• If \(c\) is negative, the graph moves downward, and each point is moved \(c\) units lower.

These shifts do not change the shape of the graph, only its position along the vertical axis. This allows us to adjust graphs to meet desired conditions without altering their fundamental form.

Horizontal Shifts

Horizontal shifts involve moving the graph of a function left or right across the x-axis. This transformation typically occurs inside the function, altering how it reacts to the input variable \(x\). For example, the function \(f(x) = \sqrt{x - c}\) is influenced in this way.

Mechanics of Horizontal Shift:

• A positive \(c\) moves the graph to the right by \(c\) units. This is because the graph is responding to values of \(x\) that are \(c\) greater than before.
• A negative \(c\) shifts the graph to the left by \(c\) units as the function adjusts to smaller \(x\) values.

These horizontal shifts allow for flexibility in aligning the curves in relation to the x-axis. It is a crucial tool for aligning graphs precisely where they need to be for specific applications or comparisons.