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Magazine Chapter 2: Problem 60
Use transformations to graph each function and state the domain and range. $$y=-\frac{1}{2} \sqrt{x+2}+4$$
Short Answer
Expert verified
Domain: \([-2, \infty)\), Range: \(( -\infty, 4 ]\)
Step by step solution
01
- Identify the base function
The base function for this transformation is the square root function: \( y = \sqrt{x} \).
02
- Horizontal shift
The expression \( x + 2 \) indicates a horizontal shift to the left by 2 units because we replace \( x \) with \( x + 2 \).
03
- Vertical compression and reflection
The coefficient \( -\frac{1}{2} \) indicates a vertical compression by a factor of \( \frac{1}{2} \) and a reflection across the x-axis. This transforms the function to \( y = -\frac{1}{2} \sqrt{x+2} \).
04
- Vertical shift
The addition of \( 4 \) to the function indicates a vertical shift upwards by 4 units. This further transforms the function to \( y = -\frac{1}{2} \sqrt{x+2} + 4 \).
05
- Determine the domain
Since the square root function is only defined for non-negative values inside the square root, \( x + 2 \geq 0 \) or \( x \geq -2 \). So, the domain is \([-2, \infty)\).
06
- Determine the range
The output of the square root function is non-negative, but it is scaled and reflected to \( y = -\frac{1}{2} \sqrt{x+2} \). Thus, the maximum value of \( -\frac{1}{2} \sqrt{x+2} \) is 0 and the minimum value is \( -\frac{1}{2} \sqrt{\infty} = -\frac{1}{2} \sqrt{\infty} = -\infty \). Adding 4 shifts the range up such that it now becomes \(( -\infty, 4 ]\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Horizontal Shift
A **horizontal shift** moves the function left or right on the grid. In the exercise, we see the transformation as \( y = -\frac{1}{2} \sqrt{x+2} + 4 \). The term \( x+2 \) indicates a horizontal shift. Since we have \( +2 \), the entire function moves left by 2 units. In general, for \( x+h \), shift left by \( h \) units. For \( x-h \), shift right by \( h \) units. Horizontal shifts only affect the input \( x \) values, but they change the location of the function on the graph.
Vertical Compression
A **vertical compression** affects the height of the graph. In the given function, \( y = -\frac{1}{2} \sqrt{x+2} + 4 \), the coefficient \( -\frac{1}{2} \) compresses the output values. Vertical compression means making the function 'shorter.' So every y-value is half its original. For \( \frac{1}{b}f(x) \), if \( b > 1 \), it compresses vertically. Example: \( y = \frac{1}{2} f(x) \) makes each y-value half the original, shrinking the graph.
Reflection
A **reflection** flips the graph over a specific axis. With \( y = -\frac{1}{2} \sqrt{x+2} + 4 \), the negative sign before \( \frac{1}{2} \) means a reflection over the x-axis. This takes each positive y-value and makes it negative. So, if \( f(x) \) becomes \( -f(x) \), all y-values flip sign. Reflections change the direction the function opens, moving above-axis parts below, and vice versa.
Vertical Shift
A **vertical shift** moves the function up or down. In \( y = -\frac{1}{2} \sqrt{x+2} + 4 \), the term \( +4 \) at the end shifts everything 4 units up. This means every y-value is increased by 4. For \( f(x) + k \), move up by \( k \) units if \( k > 0 \), and down by \( k \) units if \( k < 0 \). Vertical shifts only change the output values but retain the overall shape of the graph.
Domain and Range of Functions
The **domain** is all possible x-values for the function. Starting from the core \( y = \sqrt{x} \), it needs \( x \geq 0 \). For \( y = -\frac{1}{2} \sqrt{x+2} + 4 \), inside the root: \( x+2 \geq 0 \) means \( x \geq -2 \). So the domain is \( [-2, \infty) \). The **range** is all possible y-values. Initially, \( \sqrt{x} \) has values in \( [0, \infty) \). After transformation to \( y= -\frac{1}{2} \sqrt{x+2} \), this flips y-values to \( (-\infty, 0] \). Finally, adding 4 adjusts range to \( (-\infty, 4] \).
