Chapter 7.6, Problem 18WE

To determine

Sketch the graph of the function $f\left(x\right)=\frac{1}{4}{x}^2-2x+\frac{5}{2}$ .

Explanation of Solution

Given information: The given function is $f\left(x\right)=\frac{1}{4}{x}^2-2x+\frac{5}{2}$ .

Graph: The given function can be written as,

  $\begin{array}{c}y=\frac{1}{4}{x}^2-2x+\frac{5}{2}\mathrm{....}\left(\text{Replace }f\left(x\right)=y\right)\\ \text{or, }y-\frac{5}{2}=\frac{1}{4}\left(x^2-8x\right)\\ y-\frac{5}{2}+\left(4\right)=\frac{1}{4}\left(x^2-2\left(x\right)\left(4\right)+{\left(4\right)}^2\right)\mathrm{....}\left(\text{Complete square}\right)\\ y+\frac{3}{2}=\frac{1}{4}{\left(x-4\right)}^2\mathrm{....}\left(1\right)\end{array}$

Now (1) represents a parabola and its graph is shown as,

  

Observation: Since (1) represents a parabola of the form $y-k=a{\left(x-h\right)}^2$ . Therefore the graph for the function (1) has vertex $\left(4,-\frac{3}{2}\right)$ and axis of symmetry $x=4$ .