1.
y = x - a
reciprocal
y = 1/(x-a)
vertical when x = a
2.
y = (x-a)(x-b)
reciprocal
y = 1/[(x-a)(x-b)]
x = a and x = b
3. vertex of parabola is halfway between zeros x =(a+b)/2
2.Give a quadratic function with its zeros at x=a and x=b, what are the equations of the vertical asymptotes of its reciprocal function?
3.Once you know the asymptotes of the reciprocal of a quadratic function, how can you find its maximum or minimum?
y = x - a
reciprocal
y = 1/(x-a)
vertical when x = a
2.
y = (x-a)(x-b)
reciprocal
y = 1/[(x-a)(x-b)]
x = a and x = b
3. vertex of parabola is halfway between zeros x =(a+b)/2
A linear function in slope-intercept form is given by f(x) = mx + b, where m is the slope and b is the y-intercept. Since the zero is at x=a, we know that f(a) = 0. Substituting x=a into the equation, we get ma + b = 0. Solving for b, we find that b = -ma.
Now, the reciprocal function of f(x) is given by g(x) = 1/f(x) = 1/(mx + b).
The vertical asymptote of g(x) occurs when the denominator is equal to zero since division by zero is undefined. In this case, the denominator mx + b will be equal to zero when mx = -b. So, the equation of the vertical asymptote is mx + b = 0, which simplifies to mx = -b.
2. Similar to the previous question, let's consider a quadratic function with its zeros at x=a and x=b. The quadratic function can be written in factored form as f(x) = k(x-a)(x-b), where k is a constant.
The reciprocal function of f(x) is given by g(x) = 1/f(x) = 1/[k(x-a)(x-b)].
To find the vertical asymptotes of g(x), we need to determine when the denominator is equal to zero. Since division by zero is undefined, the denominator k(x-a)(x-b) should not be zero.
Therefore, the equations of the vertical asymptotes for the reciprocal function will be x = a and x = b. These are the values of x for which the quadratic function f(x) has zeros.
3. Once you know the asymptotes of the reciprocal of a quadratic function, you can determine its maximum or minimum by observing the behavior of the function near the asymptotes.
If the reciprocal function has a vertical asymptote at x = a, it means that as x approaches a, the function values approach infinity or negative infinity depending on the direction.
- If the function approaches positive infinity as x approaches a, it means that the quadratic function has a minimum value at x = a.
- If the function approaches negative infinity as x approaches a, it means that the quadratic function has a maximum value at x = a.
Similarly, if there is a vertical asymptote at x = b, you can determine the maximum or minimum value of the quadratic function at x = b by observing the behavior of the reciprocal function as x approaches b.
Remember, this method assumes that the reciprocal function is well-defined and behaves appropriately near the vertical asymptotes.