Linear Function Identification: Analyzing Mathematical Expressions

Linear Functions with Polynomial Forms

Determine which of the following expressions describes a linear function?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:04 Which equations represent a linear function?
00:08 Let's simplify each equation to find out.
00:12 Any equation with X to a power is not linear.
00:17 We'll apply this method to all equations: simplify and observe.
00:27 These also have X to a power, so they're not linear.
00:33 This equation fits the rules for a linear function.
00:38 And that's how we solve this problem!

Step-by-step written solution

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Understand the problem

Determine which of the following expressions describes a linear function?

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Step-by-step solution

Note that in answer A there is an exponent, therefore the answer is incorrect.

Note that in answer C, if we multiply X by X we get X to a power, therefore the answer is incorrect.

Note that in answer D there is an exponent, therefore the answer is incorrect.

In answer B the following formula can be observed.

y=mx+b y=mx+b

3

Final Answer

y=4x+1 y=4x+1

Key Points to Remember

Essential concepts to master this topic
  • Definition: Linear functions have form y = mx + b only
  • Recognition: Check highest exponent: x2 x^2 means quadratic, not linear
  • Verification: Only y=4x+1 y = 4x + 1 has degree 1 ✓

Common Mistakes

Avoid these frequent errors
  • Thinking any expression with x is linear
    Don't assume y=2x+x2 y = 2x + x^2 is linear just because it has x = wrong classification! The x2 x^2 term makes it quadratic (degree 2). Always check for the highest power of x to determine function type.

Practice Quiz

Test your knowledge with interactive questions

Which statement best describes the graph below?

xy

FAQ

Everything you need to know about this question

How do I identify if a function is linear?

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A function is linear if it can be written as y=mx+b y = mx + b where m and b are constants. The highest power of x must be exactly 1.

What if I need to expand expressions first?

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Always expand expressions like y=x(x1) y = x(x-1) to see the true form: y=x2x y = x^2 - x . The x2 x^2 term makes it quadratic, not linear.

Why isn't y=2x+x2 y = 2x + x^2 linear?

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Even though it has a linear term (2x), the x2 x^2 term makes the overall function quadratic. Linear functions can only have x to the first power.

What's the difference between linear and quadratic functions?

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Linear: highest power is x1 x^1 (straight line graph)
Quadratic: highest power is x2 x^2 (parabola graph)

Can a linear function have no x term?

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Yes! y=5 y = 5 is linear (horizontal line). It's the same as y=0x+5 y = 0x + 5 where the slope m = 0.

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