2. Explain why the graphs of reciprocals of linear functions (except horizontal ones) always have vertical asymptotes,
but the graphs of reciprocals of quadratic functions sometimes do not. (2 marks)
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Oh I'm sorry I didn't mean too, these aren't test or assignment questions they are practice test questions that I've collected and stared in my notebook for the past week from grade 12 work books. The books don't have the answers at the back and I can't find them online.
Can you please post your work and let us know where you are getting stuck? Thanks!
To understand why the graphs of reciprocals of linear functions always have vertical asymptotes, we need to first understand what a reciprocal function is. In mathematics, the reciprocal of a number is simply 1 divided by that number. Similarly, the reciprocal of a function is obtained by taking the reciprocal of the output (or y-value) of the function for each input (or x-value).
Let's start by considering a linear function of the form y = mx + b, where m and b are constants. To find the reciprocal of this function, we need to compute 1 / (mx + b) for each x-value. However, there is a vertical asymptote whenever the denominator, mx + b, equals zero since division by zero is undefined.
For a linear function, we can solve the equation mx + b = 0 to find the x-value where the denominator becomes zero. This value corresponds to the vertical asymptote of the reciprocal function. Since a linear function is a straight line, the reciprocal function will have a vertical asymptote at the x-value where the linear function crosses the x-axis.
Now, let's turn our attention to quadratic functions of the form y = ax^2 + bx + c, where a, b, and c are constants. There are cases where the reciprocal of a quadratic function does not have vertical asymptotes. This occurs when the quadratic function does not intersect or cross the x-axis.
If a quadratic function has no x-intercepts, it means the parabola is either completely above or completely below the x-axis. In this scenario, when we compute 1 / (ax^2 + bx + c), we do not encounter any x-values that result in division by zero. Hence, the reciprocal function of a quadratic function with no x-intercepts does not have vertical asymptotes.
However, when a quadratic function intersects the x-axis (meaning it has x-intercepts), division by zero can occur at the x-values where the quadratic function equals zero. These x-values correspond to the vertical asymptotes of the reciprocal function. Please note that it is possible for a quadratic function to have one, two, or no x-intercepts, resulting in different numbers of vertical asymptotes for the reciprocal function.
In summary, the graphs of reciprocals of linear functions (except horizontal ones) always have vertical asymptotes because division by zero occurs when the linear function crosses the x-axis, while the graphs of reciprocals of quadratic functions sometimes do not have vertical asymptotes when the quadratic function does not intersect the x-axis.
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