Chapter 4.1, Problem 27PPSE

Solution

To find: The vertex, axis of symmetry, maximum or minimum value and the domain and range of function $f\left(x\right)=-{\left(x-4\right)}^2-25$ .

The vertex is $\left(4,-25\right)$ . The axis of symmetry is the line $x=4$ . The function has a maximum value of $-25$ . The domain of the function is $x\in \left(-\infty ,\infty \right)$ . The range of the function is $y\le -25$ .

Given:

The function is $f\left(x\right)=-{\left(x-4\right)}^2-25$ . Re-write $f\left(x\right)=y$ gives,

  $y=-{\left(x-4\right)}^2-25\mathrm{....}\left(1\right)$

Concept used:

The vertex form of a quadratic function is $y=a{\left(x-h\right)}^2+k$ where $a\ne 0$ and the point $\left(h,k\right)$ represents the vertex of the parabola which either open upwards or downwards. The parabola opens upwards if $a>0$ , and the function has minimum value of $\left(k\right)$ and its range is all real numbers greater than or equal to $\left(k\right)$ . The parabola opens downwards if $a<0$ , and the function has a maximum value of $\left(k\right)$ and its range is all real numbers less than or equal to $\left(k\right)$ . The axis of symmetry is the line $x=h$ . And the domain of a quadratic function of the form $y=a{\left(x-h\right)}^2+k$ is all real numbers.

Calculation:

Comparing the function $\left(1\right)$ with the standard form of parabola $a=-1,h=4\text{ and }k=-25$ .

Now with the help of concept stated above:

The vertex of the function is $\left(h,k\right)=\left(4,-25\right)$ .

The axis of symmetry is the line $x=4$ .

Since $a=-1$ which is less than zero, therefore the function has maximum value of $k=-25$ and the range of the function is $y\le -25$ .

The domain of the quadratic function of the form $y=a{\left(x-h\right)}^2+k$ is all real numbers. Therefore, its domain is $\left(-\infty ,\infty \right)$ .

Conclusion:

The vertex is $\left(4,-25\right)$ . The axis of symmetry is the line $x=4$ . The function has a maximum value of $-25$ . The domain of the function is $x\in \left(-\infty ,\infty \right)$ . The range of the function is $y\le -25$ .