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Chapter 7: Problem 1

Graph each function. \(y=\sqrt{x}+1\)

Short Answer

Expert verified
The graph of the function \(y=\sqrt{x}+1\) starts at the point (0,1) and increases as \(x\) increases. The curve goes through the points (0,1), (1,2), and (4,3). The function is only defined for \(x\geq0\).

Step by step solution

01

Identify Key Points

To graph a function, it's helpful to start with identifying some key points. For the given function, \(y=\sqrt{x}+1\), one can plug in a few values of \(x\) and calculate corresponding values of \(y\). For instance, when \(x=0\), \(y=\sqrt{0}+1=1\). When \(x=1\), \(y=\sqrt{1}+1=2\). When \(x=4\), \(y=\sqrt{4}+1=3\). So, three key points on this function can be: (0,1), (1,2), and (4,3).
02

Understanding the Function's Behavior

The original square root function, \(y=\sqrt{x}\), starts at point (0,0) and increases as \(x\) increases. But for \(y=\sqrt{x}+1\), the '+1' causes every \(y\) value to get shifted up by 1. So instead of starting at point (0,0), this function starts at point (0,1).
03

Drawing the Graph

Next step is to plot these points on a graph and draw smooth curve through them. The curve should start at (0,1) and go through (1,2) and (4,3), with the end of the curve trending upwards. Because we can't take square root of negative numbers, the function is not defined for values of \(x\) less than 0. So should only be graphed for \(x\geq0\).

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Key Concepts

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

Key Points in Graphing
Graphing square root functions can be a breeze once you identify the key points. These points are pivotal because they provide a framework for sketching the entire graph. For the function \( y = \sqrt{x} + 1 \), it's beneficial to choose values of \( x \) that are easy to compute square roots for, such as 0, 1, or 4. This makes the math simpler and the graph more accurate.

Once you have selected your x-values:
  • Calculate the corresponding y-values using the function equation.
  • For \( x = 0 \), you get \( y = 1 \).
  • For \( x = 1 \), you find \( y = 2 \).
  • For \( x = 4 \), \( y \) becomes 3.
Now, you have three key points: (0,1), (1,2), and (4,3). These points will help in outlining the curve on the graph. Remember, plotting key points ensures the graph accurately represents the function.
Understanding Function Behavior
Understanding how the function behaves helps in predicting its graph. The basic square root function \( y = \sqrt{x} \) starts at the origin point (0,0) and curves upwards. However, with \( y = \sqrt{x} + 1 \), there's an important change. Every y-value is increased by 1. This means the function "lifts" the original curve upward by one unit.

In simpler terms:
  • The new starting point is (0,1) instead of (0,0).
  • The graph will trend upwards as \( x \) increases.
  • The shape remains the same, but it's elevated vertically.
These adjustments in the function's behavior are crucial. They influence how high or low the graph appears compared to its original version. Observing these changes also helps in understanding how vertical transformations alter functions in general.
Domain and Range of Functions
For any function, the domain and range describe the set of possible inputs and resulting outputs.
For the function \( y = \sqrt{x} + 1 \):
  • Domain: The values of \( x \) must be non-negative because taking the square root of a negative number isn't defined in the set of real numbers. Hence, the domain is \( x \geq 0 \).
  • Range: Since the smallest value of \( y \) occurs when \( x = 0 \) giving \( y = 1 \), this is the lowest y-value in the graph. Therefore, the range of the function is \( y \geq 1 \).
Understanding the domain and range is essential because it tells you where the function "exists" and the output values it can take. With this information, you can easily predict and graph square root functions and apply these concepts to similar transformations.

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