Chapter 2.1, Problem 14E

To determine

To sketch: The given quadratic function and compare that with the graph of function $y=x^2$ .

Explanation of Solution

Given data: $y$

The given quadratic function is $y=-x^2+2$ .

Concept used:

The standard form of quadratic function $y=a{\left(x-h\right)}^2+k$ represents a parabola with vertex is $\left(h,k\right)$ . If $a>0$ then the parabola has minima at the vertex, so concave upwards, and if $a<0$ then the parabola has maxima at the vertex so concave downwards

Calculations:

The given quadratic function is $y=-x^2+2$ may be written as

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

Now, comparing equation (1) with the function $y=a{\left(x-h\right)}^2+k$ , we have $a=-1$ , $h=0$ , and $k=2$ .

Also, for the parabola $y=x^2$ or $y=1\cdot {\left(x-0\right)}^2+0$ , we have $a=1$ , $h=0$ , and $k=0$ .

Comparison of two graphs:

Quadratic function $y=-x^2+2$ :

Since, $a=-1<0$ hence the parabola is has maximumat the vertex so is concavedown-wards, and $\left(0,2\right)$ is the vertex of the parabola. That is shown in Figure-1.

Quadratic function $y=x^2$ :

Since, $a=1>0$ hence the parabola is has minimum at the vertex so is concave upwards, and $\left(0,0\right)$ is the vertex of the parabola. That is shown in Figure-2

The sketches of the parabolas $y=-x^2+2$ and $y=x^2$ are shown here in Figure-1 and Figure-2 respectively.

  

The sktech of the graphs of functions show that the graph of the function $y=-x^2+2$ is obtained by (i) reflecting the graph of function $y=x^2$ , through x-axis, then (ii) shifting the graph by 2 unit along y-axis.