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Magazine Chapter 15: Problem 2
Explain when the power function, \(y=a x^{b},\) has only positive or only negative \(y\) -values and when it has both positive and negative \(y\) -values.
Short Answer
Expert verified
The function is only positive if \( a > 0 \) and \( b \) is even, and only negative if \( a < 0 \) and \( b \) is even. It has both values if \( b \) is odd.
Step by step solution
01
Understand the Power Function
The power function of the form \( y = ax^b \) consists of a coefficient \( a \) and an exponent \( b \). Generally, \( x \) represents the input variable, while \( y \) is the output, or result, for a given \( x \).
02
Determine the Role of Coefficient a
When \( a > 0 \), the function typically maintains the same sign as \( x^b \), depending on the value of \( b \). When \( a < 0 \), it inversely affects the sign, flipping it such that \( y \) becomes negative if \( x^b \) is positive, and positive if \( x^b \) is negative. The value of \( a = 0 \) makes the entire function \( y = 0 \), which is a special case.
03
Analyze the Power b when x is Positive or Zero
For \( x > 0 \), any positive or negative exponent \( b \) will result in positive \( x^b \) values. When \( x = 0 \), provided \( b > 0 \), \( y = 0 \). If \( b = 0 \), \( y = a \), implying \( y > 0 \) if \( a > 0 \). Hence, \( a \) affects the outcome.
04
Analyze the Power b when x is Negative
With \( x < 0 \), and \( b \) as a negative integer, \( x^b \) becomes positive. When \( b \) is an odd positive integer, \( x^b \) remains negative. If \( b \) is an even integer, \( x^b \) becomes positive. Therefore, based on the sign and parity of \( b \), \( a \) influences positivity or negativity.
05
Analyze the Function for Positive or Negative y-values
The function \( y = ax^b \) has only positive or only negative \( y \)-values if the values of \( y \) never change sign for any real \( x \). This happens if: 1) \( a > 0 \) and \( b \) is even (ensuring positivity), or 2) \( a < 0 \) and \( b \) is even (ensuring negativity).
06
Analyze the Function for Both Positive and Negative y-values
Both positive and negative \( y \)-values occur if there are changes in sign. This can occur when \( b \) is odd, leading to \( x^b \) and thus \( y \) switching sign based on the input sign of \( x \). Provided \( a eq 0 \), the function can flip from positive to negative and vice versa. This typically requires \( b \) being odd.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Coefficient impact
The coefficient, represented by the letter \( a \) in the power function \( y = ax^b \), plays a significant role in determining the sign of the output. The role of \( a \) is relatively straightforward yet impactful.
- When \( a > 0 \), the function \( y \) maintains the sign determined by \( x^b \). For example, if \( x^b \) is positive, \( y \) remains positive, and if \( x^b \) is negative, \( y \) becomes negative.
- When \( a < 0 \), the situation reverses. A negative coefficient will flip the sign of \( y \). So if \( x^b \) is positive, \( y \) will be negative, and vice versa.
- Special Case: When \( a = 0 \), the function outputs \( y = 0 \) for all values of \( x \), effectively flattening the graph into a line along the \( x \)-axis.
Exponent rules
Exponents, represented by \( b \) in the power function \( y = ax^b \), control the function's behavior with remarkable precision. The exponent not only influences the growth rate but also impacts the sign of the output.
- If \( x > 0 \), then for any real number \( b \), the value of \( x^b \) will always be positive, giving a simple predictability to the function's output.
- If \( x = 0 \) and \( b > 0 \), then \( y = 0 \), as any positive exponent will return zero. If \( b = 0 \), the function simplifies to \( y = a \), resulting in \( y \) taking the sign of \( a \).
- For negative \( x \) (i.e., \( x < 0 \)), the outcome relies heavily on whether \( b \) is odd or even: - Even exponents make \( x^b \) positive (think \( (-1)^2 = 1 \)). - Odd exponents leave \( x^b \) as negative (like \( (-1)^3 = -1 \)).
Signs of functions
The sign of a power function \( y = ax^b \) can indicate significant behavior patterns of the function. By examining the sign of \( y \), we can infer the range and nature of the output values.
- For only positive \( y \)-values, \( a > 0 \) and \( b \) must be even. This ensures that regardless of whether \( x \) is positive or negative, the function squares out to positive.
- For only negative \( y \)-values, \( a < 0 \) with even \( b \) ensures that any \( x^b \) is positive, thus flipping \( y \) to negative.
- Both positive and negative \( y \)-values exist typically if \( b \) is odd. This means the function outputs can switch from positive to negative depending on the sign of \( x \).
Odd and Even powers
The parity of exponent \( b \) in \( y = ax^b \)鈥攚hether it is odd or even鈥攑rofoundly affects the behavior and symmetry of the graph of the function.
- An even exponent \( b \) leads the power function to be symmetric about the \( y \)-axis. This symmetry means both positive and negative \( x \) values yield the same result for \( x^b \), illustrating why \( y \) will always retain the same sign, based on \( a \).
- An odd exponent \( b \) results in a function that is symmetric about the origin. As \( x \) changes sign, \( x^b \) reflects this sign change, giving \( y \) values that are both positive and negative depending on \( x \).
