PLease help!
anyy would be appreciated !thx
what is an example of a quadratic function that touches its axis and also its reciprocal?
-discuss about the vertical asymptotes of the reciprocal an why they occur?
-what can we say about the vertex of the quadratic and corresponding feature of the reciprocal?
What do you mean by "touching its axis?" Intersecting the y axis? the x axis? The axes are not properties of the function. Then you say "touching its reciprocal" do you mean the function equals its reciprocal somewhere?
Consider the function
y = (x-1)^2 -1 = x^2 -2x
It crosses the y axis (x=0) at y = 0
It crosses the x axis (y=0) at x=0 and x=2
The reciprocal is 1/[x(x-2)].
The function equals the reciprocal at
[x(x-2)]^2 = 1
x(x-2)= +1 or -1
x = 1, 1 + sqrt2
The function has a "vertex" minimum at x=1 and the reciprocal has a maximum there
Certainly! I can help you with that.
To find an example of a quadratic function that touches its axis and its reciprocal, we need to consider the general form of a quadratic function, which is f(x) = ax^2 + bx + c. For it to touch its axis, the vertex of the quadratic function needs to lie on the x-axis.
Let's start by finding the vertex of the quadratic function. The x-coordinate of the vertex can be found by using the formula x = -b / (2a). Since we want the vertex to lie on the x-axis, the y-coordinate of the vertex will be zero.
To simplify the calculation, let's assume that b = 0. This way, the quadratic function becomes f(x) = ax^2 + c. Now, we can find the x-coordinate of the vertex using x = -0 / (2a) = 0. So, the vertex is at (0, 0).
Now, let's find the reciprocal of the quadratic function. The reciprocal function is found by inverting the x and y values of the original function. Therefore, the reciprocal function of f(x) = ax^2 + c is g(x) = 1 / (ax^2 + c).
For the reciprocal function to exist and avoid division by zero, we need to make sure that the quadratic function does not have any x-values where it equals zero. These x-values are called the vertical asymptotes of the reciprocal function.
To find the vertical asymptotes, we need to solve the quadratic equation ax^2 + c = 0. Since b = 0, we can ignore the x-term. Solving for x, we get x = ± √(-c/a). If the discriminant (b^2 - 4ac) is negative, then the quadratic equation has no real solutions, which means there are no vertical asymptotes. However, if the discriminant is greater than or equal to zero, the quadratic equation has two real solutions. In this case, the quadratic function will have vertical asymptotes at those real solutions.
Now, let's consider the features of the vertex of the quadratic function and its reciprocal. The vertex of the quadratic lies on the x-axis, which means it is the lowest or highest point of the parabola depending on the coefficient a. As for its reciprocal, since it is the inverse of the quadratic function, it will have its own vertex as well. The vertex of the reciprocal function will be (0, 0) since the reciprocal of zero is still zero.
In summary, an example of a quadratic function that touches its axis and its reciprocal is f(x) = ax^2, assuming b = 0. The reciprocal of the quadratic function is g(x) = 1 / (ax^2). The quadratic function has a vertex at (0, 0), and the reciprocal function also has a vertex at (0, 0). The vertical asymptotes of the reciprocal function occur at x = ± √(-c/a) if the discriminant is greater than or equal to zero, otherwise, there will be no vertical asymptotes.