1.2: Linear and Quadratic Functions

Linear functions are those that form a line when graphed. A linear function is one of the most widely known types of functions, with the formula y=mx+b. The equation for a linear function can be found using point slope form: y-b=m(x-a) and point-point form: y-b1=m(x-a1). The slope of the line, also known as the change in y over the change in x or the rise over the run is indicated by the variable ‘m.’ A function known as a quadratic function is a polynomial. In a quadratic function f(x)=ax^2+bx+c and the graph represents a parabola, which opens either upward or downward depending on the sign of the coefficient, a. To find the roots of a function you can use either the quadratic formula or you can complete the square.

Problems:

#33: Find the roots of the quadratic polynomials.

a) 4x^2-3x-1

To find the roots of this equation I plugged the numbers into the quadratic formula, The answers that I came up with were +1 and -0.25.

b) X^2-3x-1

I solved this problem exactly the same way as the above problem. My answers were 2.41 and -0.414

#53: Show that if f(x) and g(x) are linear, then so is f(x) + g(x). Is the same true of f(x)g(x)?

The function f(x) +g(x) will still be linear if both f(x) and g(x) are linear. This is true because the addition of the two functions will still create lines, although they may be either perpendicular or parallel depending on the exact equation that defines them. F(x)g(x) will also be linear if f(x) and g(x) are independently linear. To check my assumptions, I plugged random numbers in for f(x) and g(x) and graphed them using my calculator. I also graphed f(x) + g(x) and f(x)g(x) using those numbers. The graphs for all values were horizontal lines.