Fix the mistake: Linear Functions ① Rate of Change and Graphs

Linear Functions ① Meaning and Range of a Linear Function

When y can be expressed as an equation in x and has the form y = ax + b（a ≠ 0）, y is called a linear function of x. a is called the slope, and b is called the y-intercept.

Determining Whether a Function Is Linear

When y is expressed as an equation in x, it is a linear function if it has the form y = ax + b. If a = 0, it is a constant function (a special case of direct proportion).

Example ①
Choose the functions below in which y is a linear function of x.
(A) y = 3x − 2　(B) y = x²　(C) y = 5　(D) y = −2x
Solution:(A) y=3x−2（a=3,b=−2）→linear function ✓。(D) y=−2x（a=−2,b=0）→linear function ✓。(B) is a quadratic function, and (C) is a constant function.

Range

When the range of x is determined, the range of y is also determined. If a > 0, y increases as x increases; if a < 0, the opposite is true.

Key Points
  • Form y = ax + b → linear function（a ≠ 0）
  • a: slope (when x increases by 1, y increases by a)
  • b: y-intercept (the value of y when x=0)

Practice Problems

  1. For y = −3x + 5, state the values of a and b.
  2. Express the perimeter y cm of a square with side length x cm as an equation in x, and state whether it is a linear function.

Answers and Explanations

  1. Answer:a=−3, b=5。
  2. Answer:y=4x. Since a=4,b=0, it is a linear function（y=4x+0）.

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