College Algebra, 4th Edition

Chapter 3 Review and Tests

Review

Definitions, Concepts, and Formulas

Examples

3.1 Quadratic Functions A quadratic function f is a function of the form $$f(x)=a{x}^2+bx+c,a\ne 0.$$ The standard form of a quadratic function is $$f(x)=a{(x-h)}^2+k,a\ne 0.$$ The graph of a quadratic function is a transformation of the graph of $y=x^2.$ The graph of a quadratic function is a parabola with vertex $$(h,k)=(-{\displaystyle \frac{b}{2a}},f(-{\displaystyle \frac{b}{2a}}\left)\right).$$ The maximum (if $a<0$) or minimum (if $a>0$) value of a quadratic function $f(x)=a{x}^2+bx+c$ is the y-coordinate, k, of the vertex of the parabola.

Sketch the graph of $\mathit{f}\mathbf{(}\mathit{x}\mathbf{)}\mathbf{=}\mathbf{-}\mathbf{2}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{4}\mathit{x}\mathbf{+}\mathbf{6}\mathbf{\text{.}}$ Solution We can put f into standard form: $$f(x)=-2x^2+4x+6=-2{(x-1)}^2+8.$$ The graph of f is the graph of $y=x^2$ shifted 1 unit to the right, stretched vertically by a factor ...