Chapter 2.1, Problem 38E

To determine

To write:The equation of parabola in standard form, whose graph is given, and verify the result using graphing utility.

Answer to Problem 38E

The standard equation of the graphed parabola is $f\left(x\right)={\left(x+2\right)}^2-\text{1}$ .

Explanation of Solution

Given data: $y$

The graph of the parabola is shown in Figure-1 here.

Concept used:

Everyquadratic function represents a parabola. Thestandard form ofquadratic function $f\left(x\right)=a{\left(x-\text{α}\right)}^2+\text{β}$ represents a parabola havingvertex $\left(\text{α,β}\right)$ . 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:

Let the standard quadratic function of given graphed parabola is

  $f\left(x\right)=a{\left(x-\text{α}\right)}^2+\text{β}$  .........(1)

The vertex of given parabola (from figure) is $\left(\text{α,β}\right)=\left(-2,-1\right)\Rightarrow \text{α}=-2,\text{β=-1}$

Thus, we have

  $f\left(x\right)=a{\left(x-\text{(-2)}\right)}^2+\text{(-1)}$

  $f\left(x\right)=a{\left(x+2\right)}^2-\text{1}$  .........(2)

Given parabola passes through point $\left(0,3\right)$

Therefore, from equation (2)

  $f\left(0\right)=3=a{\left(0+2\right)}^2-\text{1}$

  $\Rightarrow 4a-\text{1=3}$

  $\Rightarrow a\text{=}1$

Thus, the standard form of the given parabola is $f\left(x\right)=1\cdot {\left(x+2\right)}^2-\text{1}$

Now, the quadratic function $f\left(x\right)=1\cdot {\left(x+2\right)}^2-\text{1}$ represents given parabola with vertex $\left(-2,-1\right)$

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

The graph of the function $f\left(x\right)={\left(x+2\right)}^2-\text{1}$ (a parabola) is shown in Figure-2 here.

  

We observe that the graph of function $f\left(x\right)={\left(x+2\right)}^2-\text{1}$ is anupwards concave parabola having vertex $\left(-2,-1\right)$ , exactly same as the given graph that verify the results described above.

Conclusion:

The standard equation of the graphed parabola is $f\left(x\right)={\left(x+2\right)}^2-\text{1}$ .