Problem 30 - College Algebra

Question

Sketch a graph of the function as a transformation of the graph of one of the toolkit functions. $$m(t)=3+\sqrt{t+2}$$

Step-by-Step Solution

Answer

Shift \( \sqrt{t} \) 2 units left and 3 units up to graph \( m(t) \).

Step 1: Identify the Toolkit Function

The function given is \( m(t) = 3 + \sqrt{t+2} \). The toolkit function here is \( \sqrt{t} \), which is the square root function, a basic parent function.

Step 2: Transformation Analysis

The function \( m(t) = 3 + \sqrt{t+2} \) can be considered as a transformation of \( \sqrt{t} \). First, note the horizontal shift: replacing \( t \) with \( t+2 \) means the graph is shifted 2 units to the left. The addition of 3 indicates a vertical shift of 3 units upwards.

Step 3: Graph the Toolkit Function

Sketch the graph of the basic square root function \( \sqrt{t} \). It starts from (0, 0) and moves upwards to the right, diverging more and more slowly as \( t \) increases.

Step 4: Apply Horizontal Shift

Shift the graph of \( \sqrt{t} \) 2 units to the left. The new graph starts from the point (-2, 0) instead of (0, 0), maintaining the same shape.

Step 5: Apply Vertical Shift

Shift the graph upwards by 3 units. Thus, the starting point moves to (-2, 3), and the entire curve lifts upwards accordingly, maintaining the form of the square root function.

Step 6: Final Sketch of \( m(t) \)

Combine these transformations to get the final graph of \( m(t) = 3 + \sqrt{t+2} \). The starting point is at (-2, 3), and the graph stretches out, maintaining the trajectory of a square root function.

Key Concepts

Toolkit Functions

A toolkit function refers to a basic set of elementary functions that can be used as building blocks for more complex functions. These functions include the constant function, linear functions, quadratic functions, and others, creating a kind of "toolkit" that helps to understand transformations and relationships in algebra.
In this context, the square root function, denoted by \( \sqrt{t} \), acts as the toolkit function. It is a simple but vital function that forms the foundation for more intricate transformations. The graph of a square root function naturally increases beginning from the origin, displaying a characteristic "half-parabola" shape. Understanding these basic functions aids in predicting how they will behave under different transformations.

Square Root Function

The square root function is a fundamental element of the toolkit functions. Expressed as \( f(t) = \sqrt{t} \), its domain includes all non-negative values of \( t \), starting from zero and going to positive infinity. As \( t \) increases, the output values of \( f(t) \) increase gradually, forming a gentle curve that slopes upward to the right.
In our exercise, the square root function is central as its form is retained while undergoing transformations. The graph starts at \( (0,0) \), which is often referred to as the "parent point." It stretches with an increasing curve to the right, and this form is essential when examining how this basic graph changes due to translations and shifts.

Graph Shifts

Graph shifts are fundamental transformations that alter the position of a function's graph without changing its shape. In the given function, \( m(t) = 3 + \sqrt{t+2} \), we encounter both horizontal and vertical shifts.
Horizontal shifts occur when you adjust the function's input. Here, shifting \( t \) to \( t+2 \) moves the graph left by 2 units. This shift does not affect the basic curve of the function but merely repositions it.

• Horizontal Shift: By replacing \( t \) with \( t+2 \), the graph moves 2 units to the left.

Vertical shifts are applied by adding or subtracting directly to the function. In this case, adding 3 raises the entire graph up by 3 units.

• Vertical Shift: Adding 3 results in moving the graph up 3 units.

The graph—the square root function—maintains its shape but starts from \((-2, 3)\) instead of \((0, 0)\) due to these shifts. Knowing how shifts work allows you to manipulate and predict the graph's position effortlessly, helping you visualize complex functions by first understanding their underlying toolkit function.