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.

__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)

__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: y_{1}= 2 + √(15) and y_{2}= 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: x_{1}= 1 + √(12) and x_{2}= 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.

__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.

__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

- x^{2}+ 2 x + 3 = 0

Solutions: x_{1}= -1 and x_{2}= 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.

__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/e^{2}

- The x and y intercepts of the graph of the above equation are:

x intercepts: A = (-1+1/e^{2}, 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.

__Solution to Example 6__

- The y intercept is equal to f(0).

y = f(0) = e^{0 + 1}- 2 = e - 2

- Set y = f(x) = 0 and solve for x

e^{x + 1}- 2 = 0

e^{x + 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.

__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: x_{1}= - 1 and x_{2}= 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.

__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:

x_{1}= 7?/6 + 2k? , k=0,±1 , ±2 , ...

x_{2}= 11?/6 + 2k? , k=0,±1 , ±2 , ...

- Some of the x intercepts and the y intercept are:

x intercepts: A = (-?/6 , 0) , B = (7?/6 , 0) and C = (11?/6 , 0)

y intercepts: D = ( 0 , 1/2)

- The graph of the given function and the x and y intercepts are shown below.