Problem 35 - Thomas Calculus

Question

Graph the functions in Exercises \(35-54\) $$ y=\sqrt{x+4} $$

Step-by-Step Solution

Answer

Graph starts at (-4, 0), increasing slowly to the right.

Step 1: Identify the Function

The function we need to graph is \( y = \sqrt{x + 4} \). This is a square root function, which generally has the form \( y = \sqrt{x - h} + k \), where \( (h, k) \) represents the starting point or the vertex in a transformed scenario. For this specific equation, the function is in the form \( y = \sqrt{x + 4} \), meaning \( h = -4 \) and \( k = 0 \).

Step 2: Determine the Domain

The domain of the function \( y = \sqrt{x + 4} \) is restricted to values of \( x \) that make the expression under the square root non-negative. Therefore, solve \( x + 4 \geq 0 \), which gives \( x \geq -4 \). Hence, the domain is \( x \geq -4 \).

Step 3: Determine the Range

Since the square root function outputs only non-negative values, the range of \( y = \sqrt{x + 4} \) is all non-negative real numbers, \( y \geq 0 \).

Step 4: Plot Key Points

Calculate and plot several key points: - When \( x = -4 \), \( y = \sqrt{-4 + 4} = \sqrt{0} = 0 \) giving the point \((-4, 0)\).- When \( x = 0 \), \( y = \sqrt{0 + 4} = \sqrt{4} = 2 \) giving the point \((0, 2)\).- When \( x = 5 \), \( y = \sqrt{5 + 4} = \sqrt{9} = 3 \) giving the point \((5, 3)\).

Step 5: Draw the Graph

Using the key points you've plotted and the known shape of the square root function, draw a smooth curve starting at \( (-4, 0) \) and moving right where the curve increases slowly. Remember that the graph only exists for \( x \geq -4 \) and stays in the first quadrant since \( y \geq 0 \).

Key Concepts

Square Root Function

The square root function is a fundamental math concept often used in graphing exercises to explore curve shapes and transformations. Its standard form is \( y = \sqrt{x} \), a non-linear function creating a curve that starts from the origin and moves to the right, increasing gradually. The square root function only outputs non-negative values since a square root of a negative number is not a real number (in the context of real functions). This feature means its range starts at zero and increases. As demonstrated with \( y = \sqrt{x+4} \), there is a horizontal shift impacting the "starting point" or vertex of the graph.

Domain and Range

Understanding the domain and range of a square root function is crucial for correctly graphing it. The domain encompasses all possible input values (\(x\)) making the expression under the square root non-negative. For \( y = \sqrt{x+4} \), solve \( x + 4 \geq 0 \) to find \( x \geq -4 \). Thus, the domain is values where \( x \) are greater than or equal to \(-4\).

The range speaks to possible output values (\(y\)). Given that the square root starts at zero and grows, the range is \( y \geq 0 \). No negative values arise in the range, making square root functions synonymous with positive outputs on the graph.

Key Points Plotting

Plotting key points helps provide an accurate visual representation of the graph. Start with the vertex, where the function value moves from negative to non-negative under the square root. For \( y = \sqrt{x+4} \), the vertex is \((-4, 0)\).

Next, calculate a few strategic \( x \) values. For instance, when \( x = 0 \), plug into the function to find \( y = \sqrt{4} = 2 \), establishing \( (0, 2)\) as a point. Choose another relevant \( x \) such as \( x = 5 \), giving \( y = \sqrt{9} = 3 \). These points, once plotted, help map out the square root function's shape on a graph.

Graph Transformation

When working with transformed square root functions, adjustments like horizontal and vertical shifts can significantly affect the graph. The general form \( y = \sqrt{x-h} + k \) informs how a function shifts.

A positive or negative \( h \) shifts the graph horizontally. For \( y = \sqrt{x+4} \), \( h = -4 \), indicating a leftward move by four units. The absence of \( k \) means no vertical shift, maintaining the vertex at zero on the y-axis. Such transformations preserve the square root's intrinsic increasing curve nature but adjust the starting point, molding it to specific function characteristics.