Chapter 2.1, Problem 74E

To determine

To describe:The family of parabola represented by given function as compare with the graph of function $g\left(x\right)=x^2$ .

Explanation of Solution

Given data:

The given function is $f\left(x\right)=x^2-z$ , where z is a positive number.

Concept used:

The standard form of quadratic function $f\left(x\right)=a{\left(x-h\right)}^2+k$ represents a parabola with vertex $\left(h,k\right)$ . If $a>0$ then the graph of quadratic function opens upwards.

Comparison:

Suppose $z=3$ a positive number, then we compare parabolas $f\left(x\right)=x^2-z$ and $g\left(x\right)=x^2$ using standard equation of quadratic function and sketching the graph of functions $f\left(x\right)=x^2-2$ and $g\left(x\right)=x^2$ .

The quadratic function $f\left(x\right)=x^2-z$ can be written as

  $f\left(x\right)=1\cdot {\left(x-0\right)}^2-z$...............(1)

Thus, the function $f\left(x\right)=x^2-z$ represents a parabola with vertex $\left(0,-z\right)$ or $\left(0,-3\right)$ and concave upwards as $a=1>0$ .

The function $g\left(x\right)=1\cdot {\left(x-0\right)}^2+0$ represents a parabola with vertex $\left(0,0\right)$ and concave upwards as $a=1>0$ .

The graphs of functions $f\left(x\right)=x^2-z$ , $z=3$ and $g\left(x\right)=x^2$ are shown in Figure-1 here.

  

From the graph we see that both the functions $f\left(x\right)=x^2-z$ and $g\left(x\right)=x^2$ are concave upwards and the graph of function $f\left(x\right)=x^2-z$ is obtained by shifting the graph of function $g\left(x\right)=x^2$ through a distance z units along negative y-axis.