Problem 38 - Algebra and Trigonometry

Question

\(21-44\) . Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ y=\sqrt{x+4}-3 $$

Step-by-Step Solution

Answer

Shift the graph of \(y = \sqrt{x}\) left by 4 units and down by 3 units.

Step 1: Identify the Standard Function

The given function is based on the standard square root function, which is represented as \(y = \sqrt{x}\). This is our starting point for applying transformations.

Step 2: Identify Horizontal Shift

In the function \(y = \sqrt{x+4} - 3\), the term \(x+4\) indicates a horizontal shift. Specifically, replacing \(x\) with \(x+4\) shifts the graph of the standard function \(y = \sqrt{x}\) to the left by 4 units.

Step 3: Identify Vertical Shift

The term \(-3\) at the end of the function \(y = \sqrt{x+4} - 3\) indicates a vertical shift. This means that we take the graph of \(y = \sqrt{x+4}\) and shift it downward by 3 units.

Step 4: Sketch the Transformed Graph

Starting with the graph of \(y = \sqrt{x}\), apply the horizontal shift first by moving the entire graph left by 4 units. Then, apply the vertical shift by moving the result downward by 3 units. Ensure your new graph accurately represents these transformations.

Key Concepts

Square Root Function

The square root function is a fundamental mathematical function often represented as \(y = \sqrt{x}\). It's also one of the basic building blocks for understanding function transformations. The graph of the square root function starts at the origin point \((0, 0)\) and curves upwards and to the right in a gentle slope. This graph only exists in the first quadrant of the Cartesian plane, as square roots are typically not defined for negative numbers in basic real mathematics.

• Characteristics of this function include its increasing nature, meaning it rises from left to right.
• The domain (valid input values) is \(x \geq 0\).
• The range (output values) is \(y \geq 0\).

Understanding the base properties of the square root function is crucial before you move on to transforming it. These transformations often modify the domain, range, and position of the graph.

Horizontal Shift

A horizontal shift is one way to transform the graph of a function by moving it left or right along the x-axis. For the function given by \(y = \sqrt{x+4}\), we observe a horizontal shift. The expression inside the square root, \(x+4\), tells us that each point on the graph of the standard function \(y = \sqrt{x}\) moves to the left by 4 units.
When dealing with horizontal shifts:

• If the expression is \(x+a\), then shift left by \(a\) units.
• If the expression is \(x-a\), then shift right by \(a\) units.

In practical terms, a horizontal shift modifies the x-coordinates of every point on the graph. For example, the point \((4,0)\) on the curve \(y=\sqrt{x}\) will move to \((0,0)\) after shifting left by 4 units, effectively becoming the new starting point of the transformed graph.

Vertical Shift

A vertical shift involves moving the graph up or down along the y-axis. In the function \(y = \sqrt{x+4} - 3\), the term \(-3\) indicates a vertical shift downward by 3 units. This operation impacts the y-values for each point on the graph of the horizontally shifted square root function \(y = \sqrt{x+4}\).
To apply a vertical shift:

• Subtract \(b\) (e.g., -3) from the function to shift the graph down by \(b\) units.
• Add \(b\) to shift it up by \(b\) units.

As a result of this transformation, every point on the graph of the function \(y = \sqrt{x+4}\) has its y-coordinate reduced by 3. For instance, if a point was at \((0,0)\) after the horizontal shift, it will now be located at \((0,-3)\) after the vertical shift, creating a final transformed graph that reflects both transformations in sequence.