Which equation below represents a linear function?
[tex]y=x^2-7[/tex]
[tex]y=\frac{1}{x}+7[/tex]
[tex]y=x-7[/tex]
[tex]y=|x|+7[/tex]
Answer (2)
A linear function has the form y = m x + b .
y = x 2 − 7 is a quadratic function.
y = x 1 + 7 is a rational function.
y = ∣ x ∣ + 7 is an absolute value function.
Therefore, the linear function is y = x − 7 .
Explanation
Identifying Linear Functions We are given four equations and need to identify the one that represents a linear function. A linear function has the form y = m x + b , where m and b are constants. We will examine each equation to see if it fits this form.
Analyzing the First Equation
y = x 2 − 7 : This equation includes an x 2 term, which means it's a quadratic function, not a linear function.
Analyzing the Second Equation
y = x 1 + 7 : This equation includes a x 1 term, which means it's a rational function, not a linear function.
Analyzing the Third Equation
y = x − 7 : This equation can be written as y = 1 x + ( − 7 ) . Here, m = 1 and b = − 7 , so it fits the form y = m x + b . Therefore, this is a linear function.
Analyzing the Fourth Equation
y = ∣ x ∣ + 7 : This equation includes an absolute value term ∣ x ∣ , which means it's an absolute value function, not a linear function.
Conclusion The equation that represents a linear function is y = x − 7 .
Examples
Linear functions are used in many real-world scenarios. For example, the cost of a taxi ride can be modeled as a linear function, where the total cost is a fixed initial fee plus a rate per mile. Similarly, the relationship between temperature in Celsius and Fahrenheit is a linear function. Understanding linear functions helps us predict and analyze these types of relationships.
The equation that represents a linear function among the given options is y = x − 7 . This equation fits the standard form of a linear function, which is y = m x + b . The other equations do not match the criteria for linear functions due to their specific terms.
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