Problem 11 - A Graphical Approach to Precalculus with Limits

Question

Write an equation in \(x\) and \(y\) that results in the desired translation. Do not use a calculator. The square root function, shifted 3 units upward and 6 units to the left

Step-by-Step Solution

Answer

The translated equation is \( f(x) = \sqrt{x + 6} + 3 \).

Step 1: Identify the Original Function

The original function is the square root function, represented as \( f(x) = \sqrt{x} \).

Step 2: Translate 6 Units to the Left

To move the entire graph of a function 6 units to the left, replace \( x \) with \( x + 6 \) in the function. Now, \( f(x) = \sqrt{x} \) becomes \( f(x) = \sqrt{x + 6} \).

Step 3: Shift 3 Units Upward

To shift the graph of a function 3 units upward, add 3 to the entire function. Thus, the already-translated equation \( f(x) = \sqrt{x + 6} \) becomes \( f(x) = \sqrt{x + 6} + 3 \).

Step 4: Conclusion: Write the Translated Equation

The equation that represents the square root function shifted 3 units upward and 6 units to the left is \( f(x) = \sqrt{x + 6} + 3 \).

Key Concepts

Square Root Function

The square root function is a fundamental mathematical function that appears frequently in algebra and calculus. It is defined as \( f(x) = \sqrt{x} \) and represents the non-negative square root of \( x \). This function has a distinct shape, starting at the origin (0,0) and extending gradually in an upward curve along the positive x-axis.
Understanding the properties of the square root function is crucial. One important aspect is its domain and range. The domain of \( f(x) = \sqrt{x} \) is \( x \geq 0 \), meaning it only applies to non-negative values of \( x \). The range, or output, also comprises non-negative values. Thus, the graph of a square root function is limited to the first quadrant of the coordinate plane.

• Shape: A smooth curve beginning at (0,0)
• Domain: Values where \( x \geq 0 \)
• Range: Values where \( f(x) \geq 0 \)
• Behavior: Gradually increasing, never decreasing

Graph Transformations

Graph transformations allow us to modify the appearance or position of a graph in a coordinate plane. For the square root function, transformations include changes like shifts, reflections, stretches, and compressions. A transformation changes the function's graph by altering its equation. When applying transformations, it's important to understand the distinction between horizontal and vertical modifications:

• Horizontal Transformation: Affects the input, or \( x \)-values, thereby shifting or stretching along the x-axis.
• Vertical Transformation: Alters the output, or \( y \)-values, adjusting the graph vertically.

To perform a graph transformation, remember to adjust the function equation accordingly. For instance, a horizontal shift involves altering the \( x \) inside the function's formula, while a vertical shift modifies the entire function value.

Horizontal and Vertical Shifts

Shifting refers to moving the entire graph of a function horizontally (left or right) or vertically (up or down) without changing its shape. To shift a graph horizontally, adjust the \( x \) variable within the function:

• Move to the left: Replace \( x \) with \( x + a \) in the function, where \( a \) is the number of units to shift.
• Move to the right: Replace \( x \) with \( x - a \) in the function.

For example, to shift the square root function \( 6 \) units to the left, modify it to \( f(x) = \sqrt{x + 6} \).
Vertical shifts involve adding or subtracting a constant from the entire function value:

• Shift upward: Add a constant \( b \) to the function, \( f(x) = \sqrt{x} + b \).
• Shift downward: Subtract a constant \( b \) from the function.

Thus, adding \( 3 \) to move the function up results in \( f(x) = \sqrt{x + 6} + 3 \). These shifts allow us to relocate the graph to achieve the desired position while maintaining its original shape.