# 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. 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. 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. 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. 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. 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. 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.