FindIntercept

Middle School Number Sense Lesson 67: Finding the Intercept of a Line or Function

And now it is time for another quick geometry lesson...  By definition, the intercept of a a line is the coordinate of the point at which it crosses one of the axes.  

• The x-intercept is the point where the line crosses the x-axis (where y = 0).
• The y-intercept is the point where the line crosses the y-axis (where x = 0).

Questions about intercepts appeared 11 times last year, with a median placement at question # 67.

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You may be asked to find the intercept of a line or function in a few different ways:

• Given an equation
• Given two points
• Given an equation (missing a constant or coefficient) and the other intercept
• Given an equation of a line parallel or perpendicular to the one in question

In any of these cases, it is important to determine what the question is asking, and then plugging in information that is known (or can be quickly determined).

Definitions:

• The abscissa of a coordinate is the x-value of the ordered pair.  For example, the abscissa of (1, 2) is 1.
• The ordinate of a coordinate is the y-value of the ordered pair.  For example, the ordinate of (1, 2) is 2.
• The easiest way for me to remember which is which, is that they appear in alphabetical order.  Abscissa comes before ordinate, and x comes before y.

Example 1: If (a, b) is the y-intercept of the line 5x - 6y = 30, then b = ___

1. Since we are looking for the y-intercept, set x equal to 0.
2. 5(0) - 6y = 30 becomes -6y = 30.
3. Divide both sides by -6, and you have y = -5.

Example 2: The x-intercept of 3x - 2y = 24 is ___

1. The x-intercept means y = 0.
2. 3x - 2(0) = 24 becomes 3x = 24.
3. Divide both sides by 3, and you have x = 8.

Example 3: The abscissa of the x-intercept of the line y = .5x + 10 is ___

1. The x-intercept means y = 0.
2. 0 = .5x + 10.  Subtract 10 from both sides to get -10 = .5x
3. Divide both sides by .5 to get -20 = x.

Example 4: The ordinate of the y-intercept of the line 5y - 5x = -20 is ___

1. The y-intercept means x = 0.
2. 5y - 5(0) = -20 becomes 5y = -20.
3. Divide both sides by 5 to get y = -4.

Example 5: If the y-intercept of 4x + 3y = C is 8, then the x-intercept is ___

1. The y-intercept means x = 0.  Solve for C: 4(0) + 3(8) = C.  0 + 24 = C, so C = 24.
2. The x-intercept means y = 0.  Solve for x: 4x + 3(0) = 24.  4x = 24, so x = 6.

Example 6: If the x-intercept of 4x - 5y = k is (5, 0), then the ordinate of the y-intercept is ___

1. The x-intercept means y = 0.  Solve for k: 4(5) - 5(0) = k.  20 - 0 = k, so k = 20.
2. The y-intercept means x = 0.  Solve for y: 4(0) - 5y = 20.  -5y = 20, so y = -4.

Example 7:

The y-intercept of f(x) = 2(x - 3)² + 1 is ___
1. The y-intercept means x = 0.
2. f(x) = 2(0 - 3)² + 1 = 2(-3)² + 1 = 2(9) + 1 = 18 + 1 = 19.

Check back soon for a free worksheet to help you practice FindIntercept.

Up Next for Middle School: AddSquareDoub