For problems 7 and 8, examine each equation and determine if it represents a linear or nonlinear function. Explain your reasoning. Levi determined that y = 2 is NOT a linear functi... For problems 7 and 8, examine each equation and determine if it represents a linear or nonlinear function. Explain your reasoning. Levi determined that y = 2 is NOT a linear function because it does not take the form y = mx + b. Explain why Levi's reasoning is incorrect. How else could Levi have determined whether the graph was linear or nonlinear?
Understand the Problem
The question is asking to examine two provided equations to determine if they are linear or nonlinear functions. Furthermore, it probes into Levi's reasoning for deciding that a particular function is not linear and asks for an explanation and alternative methods for this determination.
Answer
The first equation \( y = \frac{3}{2}x + 7 \) is linear, while \( y = x^2 + 2 \) is nonlinear. Levi's claim about \( y = 2 \) is incorrect; it's linear.
Answer for screen readers
- The first equation ( y = \frac{3}{2}x + 7 ) is linear.
- The second equation ( y = x^2 + 2 ) is nonlinear.
- Levi's reasoning about ( y = 2 ) being nonlinear is incorrect because it is a linear equation ($y = mx + b$ with $m = 0$).
Steps to Solve
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Identify Linear Functions A linear function can be written in the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.
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Evaluate the First Equation The first equation is ( y = \frac{3}{2}x + 7 ). This matches the form $y = mx + b$ with $m = \frac{3}{2}$ and $b = 7$. Thus, it is a linear function.
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Evaluate the Second Equation The second equation is ( y = x^2 + 2 ). This does not match the linear form because it includes a quadratic term ($x^2$). Therefore, it is a nonlinear function.
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Analyze Levi's Reasoning Levi claims that ( y = 2 ) is not linear because it does not appear in the form ( y = mx + b ). However, ( y = 2 ) is a linear equation with $m = 0$ and $b = 2$. It represents a horizontal line.
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Alternative Methods for Determining Linearity To determine if a function is linear or nonlinear, Levi could have:
- Graphed the function: A linear function results in a straight line, while nonlinear will give curves.
- Analyzed the degree of the equation: Functions are linear if the highest exponent of $x$ is 1.
- The first equation ( y = \frac{3}{2}x + 7 ) is linear.
- The second equation ( y = x^2 + 2 ) is nonlinear.
- Levi's reasoning about ( y = 2 ) being nonlinear is incorrect because it is a linear equation ($y = mx + b$ with $m = 0$).
More Information
The function ( y = 2 ) is a special case of a linear function where the slope is zero. All lines can be characterized by their equations: vertical lines (undefined slope), horizontal lines (zero slope), and sloped lines (non-zero slope).
Tips
One common mistake is confusing constant functions with nonlinear ones. A constant function like ( y = c ) is a linear function, as it represents a horizontal line. Another mistake could be misidentifying the presence of variable exponents: only variables raised to the first power indicate a linear function.
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