Chapter 2, Problem 6CR

To determine

To compare: The graph of $y=-{\left(x+1\right)}^2$ from $y=x^2$ .

Explanation of Solution

Given:

The equation of polynomial is $y=-{\left(x+1\right)}^2$ .

Calculation:

The equations $y=x^2$ and $y=-{\left(x+1\right)}^2$ are the equation of parabola.

The graph of $y=x^2$ can be plotted as parabola with vertex at origin and open at positive y-axis which is shown below:

  

Equation of parabola is,

  $y=a{\left(x-d\right)}^2+c$

If $\left|a\right|>1$ ,

Then the graph of new parabola must be steep or narrow as compare to $y=x^2$ .

If $\left|a\right|<1$ ,

Then the graph of new parabola must be broader as compare to $y=x^2$ .

If $c<0$ ,

Then the graph will sift vertical downward by the value $c$ .

If $c>0$ ,

Then the graph will sift vertical upward by the value $c$ .

If $d>0$ ,

Then the graph will sift horizontal right by the value $d$ .

If $d<0$ ,

Then the graph will sift horizontal left by the value $d$ .

Here, $d$ is $-1$ which is $<0$ . Thus, the graph of $y={\left(x+1\right)}^2$ must be shifted horizontally leftward by the value 1 as compare to $y=x^2$ . Also, $y$ is equal to negative of $\left(x+1\right)$ which gives parabola with open end at the negative y -axes. The plot of $y=-{\left(x+1\right)}^2$ is shown below: