How to Find x and y Intercepts Of Graphs?

How to find the x and the y intercepts of graphs of functions and equations?
The x and y intercepts of a graph are points of intersection of the graph with the x axis and the y axis respectively. This a tutorial with examples and detailed solutions on how to find these points.

Example 1: Find the x and the y intercepts of the graph of function f defined by

f(x) = - 3 x + 9

Solution to Example 1

  • Since a point on the y axis has x coordinate equal to zero, to find the y interecpt, we set x to zero and find the y coordinate which is f(0).
    f(0) = -3(0) + 9 = 9
  • A point on the x axis has y coordinate equal to 0, to find the x intercept, we set y = f(x) = 0 and solve for x
    f(x) = -3 x + 9 = 0
  • Solve for x.
    x = 3
  • The x and y intercepts of the graph of f are
    x intercept: (3 , 0)
    y intercept: (0 , 9)


Example 2: Find the x and the y intercepts of the graph of the equation the circle given by

(x - 1) 2 + (y - 2) 2 = 16

Solution to Example 2

  • To find y intercept: Set x = 0 in the equation and solve for y.
    (0 - 1) 2 + (y - 2) 2 = 16
  • Solve for y
    1 + (y - 2) 2 = 16
    (y - 2) 2 = 15
    solutions: y1 = 2 + √(15) and y2 = 2 - √(15)
  • To find x intercept: set y = 0 in the given equation and solve for x
    (x - 1) 2 + (0 - 2) 2 = 16
    solutions: x1 = 1 + √(12) and x2 = 1 - √(12)
  • The x and y intercepts of the graph of the given equation are
    x intercepts: A = (1 - √(12) , 0) and B = (1 + √(12) , 0)
    y intercepts: C = (0 , 2 - √(15)) and D = (0 , 2 + √(15))
  • The graph shown below is that of the given equation. Examine the x and y intercepts and compare to those calculated. Note that the x and y intercepts may be determined graphically.
    graph of given equation example 2


Example 3: Calculate the x and the y intercepts of the graph of the linear equation given by

3x + 2y = 6

Solution to Example 3

  • Set x = 0 in the given equation and find the y intercept.
    3(0) + 2y = 6
  • Solve for y
    y = 3
  • Set y = 0 and solve for x to find the x intercept
    3 x + 2(0) = 6 , x = 2
  • The x and y intercepts of the graph of the above equation are:
    x intercepts: A = (2 , 0)
    y intercepts: B = (0 , 3)
  • The graph of the given equation is shown below. The x and y intercepts are those calculated above. Note that the x and y intercepts may be determined graphically.
    graph of given equation in example 3


Example 4: Calculate the x and the y intercepts of the graph of the quadratic function given by

f(x) = - x2 + 2 x + 3

Solution to Example 4

  • Set x = 0 in the formula of the given function and calculate the y intercept which is equal to f(0).
    y = f(0) = 3
  • To find the x intercept: set y = f(x) = 0 and solve for x
    - x2 + 2 x + 3 = 0
    Solutions: x1 = -1 and x2 = 3
  • The x and y intercepts of the graph of the above equation are:
    x intercepts: A = (-1 , 0) and B = ( 3 , 0)
    y intercepts: C = (0 , 3)
  • The graph of the given function is shown below along with the x and y intercepts as calculated above.
    graph of given equation in example 4


Example 5: Determine the x and the y intercepts of the graph of the logarithmic function given by

f(x) = - ln(x + 1) - 2

Solution to Example 5

  • Set x = 0 in the formula of the function and the y intercept is equal to f(0).
    y = f(0) = - ln(0 + 1) - 2 = - 2
  • Set y = f(x) = 0 and solve for x
    - ln(x + 1) - 2 = 0
    ln(x + 1) = -2
    x + 1 = e-2
    solution: x = - 1 + 1/e2
  • The x and y intercepts of the graph of the above equation are:
    x intercepts: A = (-1+1/e2 , 0)
    y intercepts: B = ( 0 , - 2)
  • The graph of the given function is shown below along with the x and y intercepts as calculated above.
    graph of given equation in example 5


Example 6: Calculate the x and the y intercepts of the graph of the exponential function given by

f(x) = ex + 1 - 2

Solution to Example 6

  • The y intercept is equal to f(0).
    y = f(0) = e0 + 1 - 2 = e - 2
  • Set y = f(x) = 0 and solve for x
    ex + 1 - 2 = 0
    ex + 1 = 2
    x + 1 = ln 2
    solution: x = - 1 + ln 2
  • The x and y intercepts of the graph of the above equation are:
    x intercepts: A = (-1 + ln 2 , 0)
    y intercepts: B = ( 0 , e - 2)
  • The graph of the given function and its x and y intercepts are shown below.
    graph of given equation in example 6


Example 7: Calculate the x and the y intercepts of the graph of the rational function given by

f(x) = (x 2- x - 2) / (x 2 - x - 3)

Solution to Example 7

  • The y intercept is equal to f(0).
    y = f(0) = 2/3
  • Set the numerator of f(x) equal to zero and solve for x to find the x intercepts
    x 2 - x - 2 = 0
    solution: x1 = - 1 and x2 = 2
  • The x and y intercepts of the graph of the above function are:
    x intercepts: A = (-1 , 0) and B = (2 , 0)
    y intercepts: C = ( 0 , 2/3)
  • The graph of the given function and the x and y intercepts are shown below.
    graph of given equation in example 7


Example 8: Calculate the x and the y intercepts of the graph of the sine function given by

f(x) = sin(x) + 1/2

Solution to Example 8

  • The y intercept is equal to f(0).
    y = f(0) = 1/2
  • Set f(x) equal to zero and solve for x to fnd the x intercepts
    sin(x) + 1/2 = 0 , sin(x) = -1/2
    solution:Because of the periodicity of the sine function, there is an infinite number of x intercepts given by:
    x1 = 7π/6 + 2kπ , k=0,~+mn~1 , ~+mn~2 , ...
    x2 = 11π/6 + 2kπ , k=0,~+mn~1 , ~+mn~2 , ...
  • Some of the x intercepts and the y intercept are:
    x intercepts: A = (-π/6 , 0) , B = (7π/6 , 0) and C = (11π/6 , 0)
    y intercepts: D = ( 0 , 1/2)
  • The graph of the given function and the x and y intercepts are shown below.
    graph of given equation in example 8