Investigate a quadratic which touches the x-axis and its reciprocal.
and a clear graph of the two functions on the same set of axis.
There are many such quadratic equations. Is it supposed to touch its reciprocal at the x axis, or can it be anywhere? We cannot draw graphs for you here.
To investigate a quadratic that touches the x-axis and its reciprocal, let's start by considering the general form of a quadratic function:
y = ax^2 + bx + c
We know that the quadratic touches the x-axis, so it has at least one root (x-intercept). Since the quadratic touches also its reciprocal, its graph must have symmetry about the line y = x. This implies that the x-intercepts (roots) of the quadratic will be equidistant from the line y = x.
Let's represent the x-intercepts as p and q. Since the quadratic touches the x-axis, it means that the quadratic function equals zero at these points:
0 = ap^2 + bp + c
0 = aq^2 + bq + c
To make the x-intercepts equidistant from the line y = x, we can set up the following equation:
p + q = -b/a
To create a clear graph showing both the quadratic function and its reciprocal, we can plot the points (p, 1/p) and (q, 1/q) on the graph.
Here's a step-by-step guide to finding and graphing the desired quadratic and its reciprocal:
1. Solve the system of equations:
ap^2 + bp + c = 0
aq^2 + bq + c = 0
p + q = -b/a
By solving this system, you can find the values of p, q, and ultimately, a, b, and c.
2. Once you have the values of a, b, and c, you can write down the equation of the quadratic:
y = ax^2 + bx + c
3. Calculate the reciprocal of each root:
r1 = 1/p
r2 = 1/q
4. Plot the graph of the quadratic function y = ax^2 + bx + c and its reciprocal y = 1/(ax^2 + bx + c) on the same set of axes.
Remember to carefully label the axes and indicate the points (p, 1/p) and (q, 1/q) on the graph.
By following these steps, you will be able to investigate and graph a quadratic that touches the x-axis and its reciprocal on the same set of axes.