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Magazine Chapter 1: Problem 68
Make a table of values and sketch the graph of the equation. Find the x- and y-intercepts and test for symmetry. $$y=\sqrt{x+4}$$
Short Answer
Expert verified
The graph starts at \((-4, 0)\), the x-intercept is \(-4\), the y-intercept is \(2\), and there is no symmetry.
Step by step solution
01
Create a Table of Values
First, we will create a table of values by selecting different values for \(x\) and calculating the corresponding \(y\). We choose values for \(x\) that make the expression under the square root non-negative, starting from \(-4\):\[\begin{array}{|c|c|}\hlinex & y = \sqrt{x+4} \\hline-4 & 0 \0 & 2 \5 & 3 \12 & 4 \\hline\end{array}\]
02
Plot Points and Sketch the Graph
Using the table of values, plot the points \((-4, 0)\), \((0, 2)\), \((5, 3)\), and \((12, 4)\) on a graph. Connect these points with a smooth curve to outline the graph of \(y = \sqrt{x+4}\). The graph starts at the point \((-4, 0)\) and increases to the right.
03
Identify the x-Intercept
The x-intercept is where the graph intersects the x-axis. From the table, when \(y = 0\), \(x = -4\). Therefore, the x-intercept is \(-4\).
04
Identify the y-Intercept
The y-intercept is where the graph intersects the y-axis. Substitute \(x = 0\) into the equation \(y = \sqrt{0+4} = 2\). Thus, the y-intercept is \(2\).
05
Test for Symmetry
To test for symmetry, check if the function is even, odd, or neither. An even function satisfies \(f(-x) = f(x)\) and it's symmetric about the y-axis. An odd function satisfies \(f(-x) = -f(x)\) and it's symmetric about the origin. Calculate \(f(-x) = \sqrt{-x+4}\), which is not equal to \(f(x)\), so the function is neither even nor odd and does not exhibit symmetry about the y-axis or origin.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding x- and y-Intercepts for Square Root Functions
The x- and y-intercepts of a function are crucial for understanding its graph.
These points represent where the function crosses the axes, giving us a sense of its position in the coordinate plane. For the equation \( y = \sqrt{x+4} \), let's locate these intercepts easily.
These points represent where the function crosses the axes, giving us a sense of its position in the coordinate plane. For the equation \( y = \sqrt{x+4} \), let's locate these intercepts easily.
- x-Intercept: This is where \( y = 0 \). Looking at our equation, set \( \sqrt{x+4} = 0 \). Solving this, we find \( x = -4 \). Thus, the x-intercept is at the point \((-4,0)\).
- y-Intercept: Here, we set \( x = 0 \) to find \( y \). Substitute into the equation: \( y = \sqrt{0+4} = 2 \). Therefore, the y-intercept is at the point \((0,2)\).
Exploring Symmetry in Graphs of Square Root Functions
Symmetry in graphs is a vital concept, as it can simplify your understanding of the function. A function could be symmetric about the y-axis, x-axis, or the origin. Let's explore if \( y = \sqrt{x+4} \) shows any symmetry.
- Even Functions: These satisfy \( f(-x) = f(x) \) and are symmetric around the y-axis. Checking \( \sqrt{-x+4} \) reveals it doesn't equal \( \sqrt{x+4} \), so it鈥檚 not even.
- Odd Functions: These satisfy \( f(-x) = -f(x) \) and are symmetric about the origin. Again, \( \sqrt{-x+4} \) isn't \(-\sqrt{x+4} \); thus, the function isn鈥檛 odd.
Utilizing a Table of Values to Graph Square Root Functions
Creating a table of values is a fundamental step in graphing, as it helps visualize how different values of \(x\) affect \(y\). For \( y = \sqrt{x+4} \), we select values for \(x\) ensuring the expression under the square root remains non-negative (starting at \(-4\)). Here are some considered values:
- When \( x = -4 \), \( y = 0 \)
- When \( x = 0 \), \( y = 2 \)
- When \( x = 5 \), \( y = 3 \)
- When \( x = 12 \), \( y = 4 \)
Implementing Graphing Techniques for Square Root Functions
Graphing techniques involve strategies to accurately depict a function鈥檚 graph. Let鈥檚 apply some to \( y = \sqrt{x+4} \):
- Identify the Domain: The domain is the set of viable \(x\) values, here all values \(x \geq -4\) since the square root restricts it to non-negative numbers.
- Plot Key Points: Using the table of values, plot points like \((-4,0)\), \((0,2)\), etc.
- Draw the Curve: Once points are plotted, connect them in a smooth, flowing path that reflects the increasing nature of the function as \(x\) moves away from \(-4\).
