Chapter 1.1, Problem 3GP

Solution

To graph: The function and compare that with the standard equation.

Given information:

Graph the function $f\left(x\right)=\frac{1}{2}{x}^2+2$ and compare it with the graph of $f\left(x\right)=x^2$ .

Graph:

The data points to graph the function $f\left(x\right)=\frac{1}{2}{x}^2+2$ are shown in Table -1. The graphs of the functions $f\left(x\right)=\frac{1}{2}{x}^2+2$ and $f\left(x\right)=x^2$ are shown in Fig. (1).

  

Step 1: Make a table for values of $f\left(x\right)=\frac{1}{2}{x}^2+2$ , here Table-1.

Step 2: Plot the points shown in Table-1.

Step 3: Draw smooth curve passing through the table points.

Step 4: Compare the graphs of $f\left(x\right)=\frac{1}{2}{x}^2+2$ and $f\left(x\right)=x^2$ shown in Fig. (1).

(a) The graph of the function $f\left(x\right)=\frac{1}{2}{x}^2+2$ and that of function $f\left(x\right)=x^2$ opens up.

(b) Both graphs have same axis of symmetry y -axis or line $x=0$ .

(c) Graph of function $f\left(x\right)=\frac{1}{2}{x}^2+2$ has vertex at point $\left(0,2\right)$ and the graph of function $f\left(x\right)=x^2$ at point $\left(0,0\right)$ .

(d) The distance between the vertices of graph of $f\left(x\right)=\frac{1}{2}{x}^2+2$ and graph of $f\left(x\right)=x^2$ is 2 units.

Interpretation:

The graphs of the quadratic functions are parabolas with axis symmetry $x=0$ .