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The square root function is a fundamental building block in algebra and calculus. The basic form is \( f(t) = \sqrt{t} \). It is a type of radical function and represents the operation of taking a square root. The graph of this function starts at the origin \((0,0)\) and gradually rises to the right, forming a gentle curve. This shape is because the farther you get from zero, the slower the values increase.

Key characteristics of the square root function include:

• Domain: The set of non-negative numbers \([0, \infty)\). This is because you cannot take the square root of a negative number if you're strictly dealing with real numbers.
• Range: The set of non-negative numbers \([0, \infty)\), as square roots are never negative.
• Intercept: The function passes through the origin \((0,0)\).

Understanding these properties allows you to manipulate and transform the basic square root function easily.

A horizontal shift involves moving the graph of a function to the left or right on a coordinate plane. For the square root function, a horizontal shift is introduced by changing the formula inside the square root. In the function \( m(t) = \sqrt{t + 2} \), the \(+2\) causes a horizontal shift.

How does this work? If you see \( \sqrt{t + a} \), it means the entire graph shifts left by \(a\) units. Conversely, if it were \( \sqrt{t - a} \), the graph would shift right. The intuition behind this is that by altering the input \(t\), you change where the function "starts."

In this exercise, altering from \( \sqrt{t} \) to \( \sqrt{t+2} \), shifts the graph 2 units left. This move is crucial because even a small shift can significantly impact where a graph settles on the plane.

A vertical shift translates the graph of a function up or down along the y-axis. For the function \( m(t) = 3 + \sqrt{t+2} \), the \(+3\) indicates a vertical shift upward by 3 units.

This vertical transformation is added outside the square root. It directly affects all the y-values of the function. Therefore, every point on the graph is lifted upwards by 3 units. For example, if the point \((0,0)\) on \( \sqrt{t} \) is transformed, it becomes \((0, 3)\).

The importance of vertical shifts is that they can adjust the baseline of the function without altering its shape. This aspect is incredibly helpful when solving real-world problems where an initial state or starting value must be accounted for.