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Chapter 0: Problem 40

Graph each function by plotting points and state the domain and range. If you have a graphing calculator, use it to check your results. $$ y=\sqrt{x+30} $$

Short Answer

Expert verified
Domain: \( [-30, \, \infty) \); Range: \( [0, \, \infty) \). Plot points and draw the graph.

Step by step solution

01

- Understand the Function

The given function is \( y = \sqrt{x+30} \). This means \( y \) is the square root of \( x+30 \).
02

- Identify the Domain

To find the domain, set the expression inside the square root to be greater than or equal to zero: \( x + 30 \geq 0 \). Solve for \( x \): \( x \geq -30 \). So, the domain is \( [-30, \, \infty) \).
03

- Identify the Range

The square root function result is always non-negative. Thus, the range of \( y = \sqrt{x+30} \) is all non-negative real numbers: \( [0, \, \infty) \).
04

- Plot Points

Select a few points for \( x \) and compute \( y \): - For \( x = -30 \): \( y = \sqrt{-30 + 30} = 0 \) - For \( x = -20 \): \( y = \sqrt{-20 + 30} = \sqrt{10} \approx 3.16 \) - For \( x = -10 \): \( y = \sqrt{-10 + 30} = \sqrt{20} \approx 4.47 \) - For \( x = 0 \): \( y = \sqrt{0 + 30} = \sqrt{30} \approx 5.48 \)
05

- Draw the Graph

Plot the points from Step 4 on a coordinate plane and connect them with a smooth curve. The graph starts at \( (-30, 0) \) and moves upward to the right.
06

- Check with Graphing Calculator

Use a graphing calculator to plot \( y = \sqrt{x+30} \) and verify that the manually drawn graph is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Domain and Range
Understanding the domain and range of a function is crucial for graphing. For the function \( y = \sqrt{x+30} \), the domain is where the function is defined. Because you can't take the square root of a negative number in the set of real numbers, you need \( x + 30 \geq 0 \). Solving this, we get that \( x \) must be greater than or equal to -30. So, the domain of this function is \( [-30, \infty) \).

The range is all possible \( y \)-values. Since the square root function gives non-negative results, \( y \) will always be 0 or positive. So, the range of \( y = \sqrt{x+30} \) is \( [0, \infty) \). This means our graph will start at the point (-30,0) and increase without bound.
Plotting Points
Plotting points helps us visualize the function. Start by selecting \( x \) values within our domain. Let's choose a few key points:

  • When \( x = -30 \), \( y = \sqrt{-30+30} = 0 \)
  • When \( x = -20 \), \( y = \sqrt{-20+30} \approx 3.16 \)
  • When \( x = -10 \), \( y = \sqrt{-10+30} \approx 4.47 \)
  • When \( x = 0 \), \( y = \sqrt{0+30} \approx 5.48 \)
Plot each of these points on the coordinate plane. This will give you:

- (-30, 0)
- (-20, 3.16)
- (-10, 4.47)
- (0, 5.48)

Connect these points smoothly. The graph will start at (-30, 0) and then rise to the right.
Graphing Calculator Verification
Using a graphing calculator can help to check your work.

1. Enter the function \( y = \sqrt{x + 30} \) into the calculator.
2. Verify that the graph matches your manually plotted points.
3. Look at how the graph behaves鈥攕tarting from the point (-30,0), it should rise smoothly upwards to the right.

Graphing calculators often provide a more precise visualization. It's a great tool, especially to confirm your manually drawn graphs.

If you find any discrepancies, double-check your plotted points and calculations to correct any mistakes.

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