Question

the graph of a function must be linear if it has what characteristic?

it passes through the origin.
it crosses the x-axis more than once.
it crosses the y-axis exactly once.
it has a constant slope.

Explanation:

Analyze the definition of a linear function

A function is linear if and only if its rate of change (slope) is constant throughout its entire domain.

Evaluate the given options

• "It passes through the origin": Non-linear functions like \(y = x^2\) also pass through the origin.
• "It crosses the x-axis more than once": A non-zero linear function can cross the x-axis at most once.
• "It crosses the y-axis exactly once": All functions must cross the y-axis at most once to satisfy the vertical line test, including non-linear ones like \(y = x^2 + 1\).
• "It has a constant slope": This is the defining characteristic of a linear function.

Answer:

• It passes through the origin.
• It crosses the x-axis more than once.
• It crosses the y-axis exactly once.
• It has a constant slope. (Correct answer)