how to find x intercept of a function

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 is 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

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

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

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

Example 6

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

f(x) = e^{x + 1} - 2

Solution to Example 6

Example 7

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

f(x) = (x²- x - 2) / (x² - x - 3)

Solution to 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.