Chapter 8, Problem 12CT

To determine

Tofind:The quadratic function in standard form whose graph satisfies the given conditions.

Answer to Problem 12CT

The quadratic function of even degree and with range $y\ge 3$ is $f\left(x\right)=x^2+3$ .

Explanation of Solution

Given information:

Condition: The quadratic function is of even degree and its range is $y\ge 3$ .

Formula used:

The standard form of quadratic function is written as $f\left(x\right)=a{\left(x-h\right)}^2+k$ having vertex at point $\left(h,k\right)$ .The range of $f\left(x\right)$ is $y\ge k$ .

Calculations:

Let $f\left(x\right)=a{\left(x-h\right)}^2+k$ be the quadratic function of even degree and its range is $y\ge 3$ .

The function is of even degree if $h=0$ and its range $y\ge 3$ only when $k=3$ . Therefore, function is

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

  $f\left(x\right)=a{x}^2+3$

Since, range of $f\left(x\right)$ is $y\ge 3$ , therefore, the function $f\left(x\right)$ opens up, i.e., $a>0$ .

For simplicity, we take $a=1$ , then the quadratic function of even degree and with range $y\ge 3$ is $f\left(x\right)=1\cdot x^2+3$ Or $f\left(x\right)=x^2+3$ .

Conclusion:

Thequadratic function of even degree and with range $y\ge 3$ is $f\left(x\right)=x^2+3$ .