How does the slope of a linear function affect the number of zeros of that function?

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Understanding Slope and Zeros of Linear Functions

A linear function can be represented in the form of y = mx + b, where m is the slope and b is the y-intercept. The slope indicates how steep the line is and the direction in which it moves—upward if positive and downward if negative.

Relationship Between Slope and Zeros

The extbf{zero} of a function, also known as the extbf{x-intercept}, refers to the value of x when y = 0. To find the zeros of a linear function, you set the equation to zero and solve for x:

0 = mx + b

Proceeding with the calculation, we can isolate x:

mx = -b
x = -b/m

From this equation, we can conclude that:

  • If the slope (m) is not equal to zero (m ≠ 0), there will be precisely one zero (one x-intercept) on the graph of the linear function as the line will intersect the x-axis at a single point.
  • If the slope (m) is zero (m = 0), the equation becomes y = b, which indicates a horizontal line. In this case, the function will either have no zeros if b is not equal to zero (the line does not touch the x-axis) or infinitely many zeros if b = 0 (the line lies on the x-axis).

Conclusion

In summary, the slope of a linear function plays a critical role in determining the number of zeros or x-intercepts. A non-zero slope guarantees one x-intercept, whereas a zero slope leads to either none or infinitely many, depending on the y-intercept.

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