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OCF.02.6 - Reciprocals of Quadratic Functions

OCF.02.6 - Reciprocals of Quadratic Functions. MCR3U - Santowski. (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}

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OCF.02.6 - Reciprocals of Quadratic Functions

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

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

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

  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. (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. (F) Homework • Handout

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