1. 1 OCF.02.6 - Reciprocals of Quadratic Functions MCR3U - Santowski

2. 2 (A) Review of the Reciprocal Function We have seen what the reciprocal function f(x) = 1/x looks like: Its domain is {x E R| x  0} Its range likewise is {y E R| y  0} We have a vertical asymptote on the y-axis (x = 0) and and horizontal asymptotes on the x-axis (y = 0) Two key points are (1,1) and (-1,-1)

3. 3 (B) Reciprocals of Linear Functions Now if we take any linear function (y = mx + b), we can graph its reciprocal The x-intercept on the linear function is –b/m and this is where we find the vertical asymptote The horizontal asymptote remains on the x-axis

4. 4 (C) Reciprocal of Quadratic Functions Now we apply the same idea of a reciprocal to quadratic functions We have seen that the key points on a function are the x-intercepts (as these form the vertical asymptotes of the reciprocal function) and we know that quadratic functions have either 0,1, or 2 x-intercepts Therefore, we expect the reciprocal function to have either 0,1, or 2 vertical asymptotes The horizontal asymptote will remain y = 0, as our reciprocal function equation is 1/f(x) so as f(x) gets larger, the value of the reciprocal gets smaller

5. 5 (D) Graphs of Reciprocal Quadratic Functions Here are some graphs of quadratic functions that have 2 x-intercepts and their reciprocal function:

6. 6 (D) Graphs of Reciprocal Quadratic Functions Here are some graphs of quadratic functions that have 1 x-intercept and their reciprocal function:

7. 7 (D) Graphs of Reciprocal Quadratic Functions Here are some graphs of some quadratic functions that have no x-intercepts and their the reciprocal function:

8. 8 (E) Analysis of the Reciprocal of Quadratic Functions The domain is restricted to wherever the x-intercept(s) of the original quadratic function A key point on the original was the vertex  in the reciprocal, the vertex relates to a range restriction  in that the reciprocal of the y- value of the vertex is a key point on the reciprocal The vertical asymptote(s) occur where the roots of the original were The horizontal asymptote is y = 0 or the x-axis

9. 9 (F) Homework Handout