Problem 18 - Algebra 2

Question

Graph each function. State the domain and range of each function. \(y=5-\sqrt{x+4}\)

Step-by-Step Solution

Answer

Domain: [-4, ∞); Range: (-∞, 5]

Step 1: Identify the Function Type

The given function is \( y = 5 - \sqrt{x+4} \). This is a transformed square root function, which involves a vertical shift and reflection.

Step 2: Determine the Domain

The square root function \( \sqrt{x+4} \) is only defined for \( x+4 \geq 0 \), which means \( x \geq -4 \). Therefore, the domain in interval notation is \([-4, \infty)\).

Step 3: Determine the Range

Since \( y = 5 - \sqrt{x+4} \), the maximum value of \( \sqrt{x+4} \) is 0 (at \( x = -4 \)) and decreases from there as \( x \) increases. Thus, \( y \) has a maximum value of 5. The range of the function is \((-fty, 5] \).

Step 4: Sketch the Function

Start by plotting the key point \((-4, 5)\) since when \( x = -4 \), \( y \) is 5. As \( x \) increases, plot additional points such as when \( x = 0, y = 3 \), building a curve extending downward to the right. The curve reflects over the horizontal line through \( y = 5 \).

Step 5: Conclude with the Full Graph

Draw a smooth curve gradually decreasing from the point \((-4, 5)\) through the plotted points, ensuring the graph does not fall below 5 anywhere across its domain. This completes the graph of \( y = 5 - \sqrt{x+4} \).

Key Concepts

Domain of a Function

When considering a function like \( y = 5 - \sqrt{x+4} \), the domain is all the permissible values of \( x \). To determine the domain for this specific function, you must consider the expression inside the square root: \( \sqrt{x+4} \).

This part of the mathematical expression is only defined when \( x+4 \geq 0 \). Thus, we find \( x \geq -4 \). This means any value of \( x < -4 \) would result in taking the square root of a negative number, which isn't possible with real numbers. Therefore, the domain of the function is restricted to:

• \([-4, \infty)\) in interval notation.

This tells us \( x \) can be as small as \(-4 \), going infinitely in the positive direction.

Range of a Function

The range of a function is all the possible values that \( y \) can take. For our transformed square root function \( y = 5 - \sqrt{x+4} \), understanding how the function behaves helps us determine its range.

Initially, consider the square root component \( \sqrt{x+4} \). As we've learned earlier, this expression is defined for \( x > -4 \). The value of \( \sqrt{x+4} \) starts at \( 0 \) when \( x = -4 \) and increases as \( x \) becomes larger. However, since our whole function is \( 5 - \sqrt{x+4} \), it means the function starts at a maximum of \( 5 \) when \( x \) is \(-4\).

As \( x \) increases, \( \sqrt{x+4} \) becomes larger, making the whole \( 5 - \sqrt{x+4} \) expression smaller. This demonstrates that as \( x \) becomes large, \( y \) decreases continuously, but it never goes below any particular value since \( \sqrt{x+4} \) grows indefinitely.

Thus, the range is:

• \(( -\infty, 5 ]\)

\( y \) can take any value less than or equal to 5.

Graphing Functions

Graphing this function, \( y = 5 - \sqrt{x+4} \), involves plotting points and connecting them to reveal the general shape of the graph. Begin by identifying important points that serve as a guide for drawing this curve.

The key starting point, also considered the vertex in this context, is \( (-4, 5) \). Here, at \( x = -4 \), the value of \( y \) is at its maximum, \( 5 \). As \( x \) increases from this point, calculate additional points to assist in sketching: for instance, at \( x=0 \), we find \( y=3 \).

Mark these on your graph, and draw a curve connecting them, starting from the vertex downward to the right. The curve should gradually decrease smoothly, reflecting vertically from a horizontal line at \( y = 5 \).

Remember:

• The graph never goes above \( y = 5 \)
• The slope is always descending as \( x \) increases

By following these steps, you effectively plot the transformed square root function on a coordinate plane, visualizing its domain and range.