A discussion of what happens when a linear function is equal to a quadratic function

By Michelle Jones and Vicki Tarleton

While working on our instructional unit, we came up with the title for a potential lesson, "When linear meets quadratic." Dr. Wilson saw the title and asked, "What happens when linear equals quadratic?" He proceeded to write ax^2 +bx +c = xy in our notes. Indeed, this made us curious. So let the investigation begin!

First we examined when a=1, b=1, and c=1

Immediate observations:

There are two branches: the result of a rational function.

The y-axis is a vertical asymptote which means that as x approaches 0, the function is undefined.

There is a slant asymptote of y = x which means the degree of the polynomial of the numerator is greater than the ratio of the degree of the denominator. Furthermore, the ratio of the leading coefficients of the second degree term to the first degree term is 1.

Examining how "a" affects the graph:

 Positive "a" value "a" is zero Negative "a" value a = 3 a = 0 a = -3 Slant asymptote y = 3x +1 Horizontal asymptote y = 1 x = 0 Slant Asymptote y = -3x + 1

Observations:

"a" controls the width of the branches and determines the types and location of the asymptotes

Examining how "b" affects the graph.

 Positive "b" value "b" is Zero Negative "b" value b=3 b=0 b=-3 Vertical shift up 3 units No change in original graph Vertical shift down 3 units

Observations:

"b" controls the vertical shift of the function

Examining how "c" affects the graph

 Positive "c" value "c" value of zero Negative "c" value c = 3 c = 0 c = -3 Branches occupy Quadrants 1 and 3 Vertical Asymptote x = 0 Slant Asymptote y = x + 1 Branches collape to slant aymptote y = x +1 Branches occupy Quadrants !,4 and 2,3 Vertical asymptote x=0 Slant asymptote y = x + 1

Observations:

The vertical and slant asymptotes do not change, but the branches' direction of opening does.

Conclusion

Rational Functions can be defined as the quotient of two polynomials. It has the form f(x) = g(x) / h(x),

where h(x) is not equal to zero. Therefore, what we started with was a rational function written in another form:

f(x)*h(x) = g(x), where h(x) is linear and g(x) is quadratic.

The branches of a rational function approach lines called asymptotes.

If the function is not defined when x = a, then the line with the equation x = a is a vertical asymptote

The line y = b is a horizontal asymptote for a function f(x) if f(x)---> b as x--->infinity or as x--->negative infinity.

The oblique line "l" is a slant asymptote for a function f(x) if the graph of f(x) approaches "l" as x---> infinity or as x---> negative infinity.