Problem 41 - Intermediate Algebra

Question

Graph each of the functions. $$f(x)=-2 \sqrt{x+3}+4$$

Step-by-Step Solution

Answer

Graph starts at (-3, 4), goes down following transformations for square root function.

Step 1: Identify the Function Type

The given function is a transformed square root function of the form \(f(x) = -2 \sqrt{x+3} + 4\). It involves transformations of the basic square root function \(\sqrt{x}\).

Step 2: Determine the Domain

In the square root function \(\sqrt{x+3}\), set \(x+3 \geq 0\) to avoid taking the square root of a negative number. Solving \(x+3 \geq 0\) gives \(x \geq -3\). Thus, the domain of \(f(x)\) is \([-3, \infty)\).

Step 3: Identify Function Transformations

The function has several transformations: a horizontal shift 3 units to the left, a vertical stretch by a factor of -2 (which also reflects in the x-axis), and a vertical shift 4 units up. This means you start with the basic graph \(y = \sqrt{x}\), shift it left by 3 units, reflect it across the x-axis and stretch by a factor of 2, then shift it up by 4 units.

Step 4: Plot Key Points

Use transformations to determine new key points. Start with \((0, \sqrt{0})\) from the parent function, apply transformations to plot \((-3, 4)\). Use other points like \((-2, \sqrt{1})\) becoming \((-2, 2)\) due to transformations, then plot it. Calculate \((-1, \sqrt{2})\) to \((-1, 1.17)\) and plot it.

Step 5: Graph the Function

Begin by marking the points and the domain on a set of axes. Sketch the graph starting from \((-3, 4)\), incorporating the transformations. The graph should be a downward curve starting at \((-3, 4)\) and going through the plotted points, with the direction determined by the reflection in the x-axis.

Key Concepts

Square Root Functions

A square root function is one of the most fundamental functions in algebra. It is expressed in the form \(y = \sqrt{x}\). This function only takes non-negative numbers under the square root to produce a real output. Since square roots of negative numbers do not yield real numbers, the typical domain of a simple square root function is all non-negative numbers.In the context of more complex functions, like \(f(x) = -2 \sqrt{x+3} + 4\), the expression inside the square root \(x+3\) can shift the domain:

• Determine when \(x+3 \geq 0\) to ensure all inputs are valid.
• This results in \(x \geq -3\), setting the domain to \([-3, \infty)\).

By understanding the properties of the basic square root function, graphing becomes simpler even with transformations applied.

Function Transformations

Function transformations are operations that change the position or shape of a graph in a coordinate plane. There are several types of transformations, and understanding them is crucial in graphing functions like \(f(x) = -2 \sqrt{x+3} + 4\).Let's break it down:

• Horizontal Shifts: Adding or subtracting a number inside the square root moves the graph left or right. For example, \(\sqrt{x+3}\) shifts the graph 3 units to the left, since you add 3 inside the square root.
• Vertical Stretches/Reflections: Multiplying the square root by a negative or positive number can stretch and reflect the graph. Here, \(-2 \sqrt{x+3}\) means stretching the graph by a factor of 2 and reflecting it across the x-axis.
• Vertical Shifts: Adding or subtracting a number outside the square root moves the graph up or down. Adding 4 means that every point is moved up 4 units.

Using these transformations makes it easier to move from the basic graph of \(y = \sqrt{x}\) to any transformed version.

Domain of a Function

The domain of a function includes all possible input values that will yield a real output. In square root functions, it's particularly important to consider the values that prevent operations like taking the square root of a negative number.For a function such as \(f(x) = -2 \sqrt{x+3} + 4\), the domain is determined by the expression under the square root. Here's how you find the domain:

• Gather expressions like \(x+3\) that must be non-negative.
• Set up an inequality: \(x+3 \geq 0\).
• Solve for \(x\) to find \(x \geq -3\).

This condition ensures that every input in the domain leads to a real output. Therefore, the domain of this function is \([-3, \infty)\), accommodating all transformations applied to the square root.