Chapter 2.1, Problem 7E

To determine

To find:The quadratic function with its graph.

Answer to Problem 7E

The graph (b) is the graph of quadratic function $f\left(x\right)=x^2+3$ .

Explanation of Solution

Given data: $y$

The given quadratic function is $f\left(x\right)=x^2+3$ . The graphs are shown in figure here.

  

Method/Formula used:

The standard form of quadratic function $f\left(x\right)=a{\left(x-h\right)}^2+k$ represents a parabola whose 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

Calculation:

The given quadratic function is $f\left(x\right)=x^2+3$ may be written as

  $f\left(x\right)=1\cdot {\left(x-0\right)}^2+3$  .........(1)

Now, comparing equation (1) with the function $f\left(x\right)=a{\left(x-h\right)}^2+k$ , we have $a=1$ , $h=0$ , and $k=3$ .

Since, $a=1>0$ hence the parabola is has minimum at the vertex so is concave upwards.

Thus, the quadratic function $f\left(x\right)=x^2+3$ is the equation of a parabola that is concave upwardswith vertex $\left(0,3\right)$ that matches with the graph (b).

Hence, the graph (b) is the graph of quadratic function $f\left(x\right)=x^2+3$

Conclusion:

The graph (b) is the graph of quadratic function $f\left(x\right)=x^2+3$ .