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Concept Version 9
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Zeroes of Linear Functions

A zero, or xxx-intercept, is the point at which a linear function's value will equal zero.

Learning Objective

  • Practice finding the zeros of linear functions


Key Points

    • A zero is a point at which a function's value will be equal to zero. Its coordinates are (x,0)(x,0)(x,0), where xxx is equal to the zero of the graph.
    • Zeros can be observed graphically or solved for algebraically.
    • A linear function can have none, one, or infinitely many zeros. If the function is a horizontal line (slope = 000), it will have no zeros unless its equation is y=0y=0y=0, in which case it will have infinitely many. If the line is non-horizontal, it will have one zero.

Terms

  • y-intercept

    A point at which a line crosses the yyy-axis of a Cartesian grid.

  • zero

    Also known as a root; an xxx value at which the function of xxx is equal to 000.

  • linear function

    An algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable. 


Full Text

The graph of a linear function is a straight line. Graphically, where the line crosses the xxx-axis, is called a zero, or root.  Algebraically, a zero is an xxx value at which the function of xxx is equal to 000.  Linear functions can have none, one, or infinitely many zeros.  If there is a horizontal line through any point on the yyy-axis, other than at zero, there are no zeros, since the line will never cross the xxx-axis.  If the horizontal line overlaps the xxx-axis, (goes through the yyy-axis at zero) then there are infinitely many zeros, since the line intersects the xxx-axis multiple times.  Finally, if the line is vertical or has a slope, then there will be only one zero.  

Finding the Zeros of Linear Functions Graphically

Zeros can be observed graphically.  An xxx-intercept, or zero, is a property of many functions. Because the xxx-intercept (zero) is a point at which the function crosses the xxx-axis, it will have the value (x,0)(x,0)(x,0), where xxx is the zero.

All lines, with a value for the slope, will have one zero.  To find the zero of a linear function, simply find the point where the line crosses the xxx-axis. 

Zeros of linear functions

The blue line, y=12x+2y=\frac{1}{2}x+2y=​2​​1​​x+2, has a zero at (−4,0)(-4,0)(−4,0); the red line, y=−x+5y=-x+5y=−x+5, has a zero at (5,0)(5,0)(5,0).  Since each line has a value for the slope, each line has exactly one zero.  

Finding the Zeros of Linear Functions Algebraically

To find the zero of a linear function algebraically, set y=0y=0y=0 and solve for xxx. 

The zero from solving the linear function above graphically must match solving the same function algebraically.  

Example: Find the zero of y=12x+2y=\frac{1}{2}x+2y=​2​​1​​x+2 algebraically 

First, substitute 000 for yyy: 

0=12x+2\displaystyle 0=\frac{1}{2}x+20=​2​​1​​x+2 

Next, solve for xxx. Subtract 222 and then multiply by 222, to obtain: 

12x=−2x=−4\displaystyle \begin{aligned} \frac{1}{2}x&=-2\\ x&=-4 \end{aligned}​​2​​1​​x​x​​​=−2​=−4​​ 

The zero is (−4,0)(-4,0)(−4,0).  This is the same zero that was found using the graphing method.  

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