Chapter 8, Problem 6CT

To determine

To graph: The quadratic function $y=-\left(x+5\right)\left(x-1\right)$ and evaluate the domain and range of the function.

Explanation of Solution

Given information:

The equation of quadratic function $y=-\left(x+5\right)\left(x-1\right)$ .

Graph:

Consider the provided function $y=-\left(x+5\right)\left(x-1\right)$ .

Expand the function,

  $\begin{array}{c}y=-\left(x+5\right)\left(x-1\right)\\ =-\left[x^2-x+5x-5\right]\\ =-\left[x^2+4x-5\right]\\ =-x^2-4x+5\end{array}$

The function is of the form $f\left(x\right)=a{x}^2+bx+c$ where a is negative. Here $a=-1$ , the function is a parabola that open downward since $a<0$ . Also, the c represents the y -intercept of the parabola.

Therefore, the graph touches the y -axis at the point $\left(0,5\right)$ .

Now, factor form of the function is $y=-\left(x+5\right)\left(x-1\right)$

Now, the zeroes of the function is when $f\left(x\right)$ is equated to 0.

Therefore, $-\left(x+5\right)\left(x-1\right)=0$

Now, the zeroes of polynomial are $-5\text{ and }1$ . So, the parabola passes through the points $\left(-5,0\right)\text{ and }\left(1,0\right)$ .

Plot the ordered pairs on a coordinate plane and join the points to obtain the smooth curve.

The red curve represents the quadratic function $y=-\left(x+5\right)\left(x-1\right)$ . Also the vertex of the graph is at the point $\left(-2,9\right)$ which is the maximum point of the graph.

  

Interpretation:

Domain of the function is the set of values of independent variable for which function is defined.

The range of the function is the set of values of the dependent variable.

The quadratic functions $y=-\left(x+5\right)\left(x-1\right)$ is a polynomial function and polynomial functions are defined on the entire number line. Therefore, domain of the function is $\left(-\infty ,\infty \right)$ . Now, the range of the function is the set of all values of y such that y is less than nine. The same can be observed from the graph. Therefore, range of the function is $\left(-\infty ,9\right]$ .