Question

What is the equation of a linearized quadratic equation?

What is the equation of a linearized inverse equation?

Answer 1

A)Equation of a linear quadratic equation

Ans:

A quadratic equation is defined as an equation in which one or more of the terms is squared but raised to no higher power. The general form is ax² + bx + c = 0, where a, b and c are constants.

In a linear- quadratic system where only one variable in the quadratic is squared, the graphs will be a parabola and a straight line. When graphing a parabola and a straight line on the same set of axes, three situations are possible.

The equations will intersect in two locations. Two real solutions.	The equations will intersect in one location. One real solution.	The equations will not intersect. No real solutions.

We have probably solved systems of linear equations. But for a system of two equations where one equation is linear, and the other is quadratic then,

We can use a version of the substitution method to solve systems of this type.

Remember that the slope-intercept form of the equation for a line is y=mx+by=mx+b, and the standard form of the equation for a parabola with a vertical axis of symmetry is y=ax2+bx+c,  a≠0y=ax2+bx+c,  a≠0.

To avoid confusion with the variables, let us write the linear equation as y=mx+dy=mx+d where mm is the slope and dd is the yy-intercept of the line.

Substitute the expression for yy from the linear equation, in the quadratic equation. That is, substitute mx+dmx+d  for yy in y=ax2+bx+cy=ax2+bx+c .

mx+d=ax2+bx+cmx+d=ax2+bx+c

Now, rewrite the new quadratic equation in standard form.

Subtract mx+dmx+d  from both sides.

(mx+d)−(mx+d)=(ax2+bx+c)−(mx+d)0=ax2+(b−m)x+(c−d)(mx+d)−(mx+d)=(ax2+bx+c)−(mx+d)0=ax2+(b−m)x+(c−d)

Now we have a quadratic equation in one variable, the solution of which can be found using the quadratic formula.

The solutions to the equation ax2+(b−m)x+(c−d)=0ax2+(b−m)x+(c−d)=0   will give the xx-coordinates of the points of intersection of the graphs of the line and the parabola. The corresponding yy-coordinates can be found using the linear equation.

Another way of solving the system is to graph the two functions on the same coordinate plane and identify the points of intersection.

B) Equation of linearised inverse equation

Ans:

The inverse of a linear function is much easier to find as compared to other kinds of functions such as quadratic and rational. The reason is that the domain and range of a linear function naturally span all real numbers unless the domain is restricted.

Steps in Finding the Inverse of a Linear Function

1. Replace f(x) by y
2. Switch the roles of “x” and “y”, in other words, interchange x and y in the equation.
3. Solve for y in terms of x
4. Replace y by f −1(x) to get the inverse function
5. Sometimes, it is helpful to use the domain and range of the original function to identify the correct inverse function out of two possibilities. This happens when you get a “plus or minus” case in the end.

We will consider an example

How to Straighten an Inverse quadratic curve

An inverse curve is a curve of the general form y = (a/x) + b, where a and b are constants or coefficients. An inverse curve can be plotted as a straight line, which has the general form y = mx + c, where m is the gradient and c is the y-intercept, by calculating the inverse or "reciprocal" of the x coordinates and then replotting them against the original y coordinates. You can straighten a curve to easily determine the coefficients of the inverse curve.

Message: I have tried to give equations for some simple systems as you have not mentioned the appropriate system to be evaluated.