Chapter 4, Problem 5CT

Solution

To identify the axis of symmetry, maximum & minimum values, and domain & range.

Axis of symmetry: $x=2.5$

Vertex: $(2.5,3.25)$

Domain: $(-\infty ,\infty )$

Range: $(-\infty ,3.25]$

Given:

The function is given by:

  $y=-x^2+5x-3$

Formula used:

The axis of symmetry for the parabola with the equation of the form $y=a{x}^2+bx+c$ is given by:

  $x=-\frac{b}{2a}$

Calculation:

Comparing the given function with the parabola equation of the form $y=a{x}^2+bx+c$ , the parameters are:

  $\begin{array}{l}a=-1\\ b=5\\ c=-3\end{array}$

Now, the axis of symmetry is given by:

  $\begin{array}{l}x=-\frac{b}{2a}\\ \Rightarrow x=-\left(\frac{\left(5\right)}{2\left(-1\right)}\right)\\ \Rightarrow x=\frac{5}{2}\\ \Rightarrow x=2.5\end{array}$

Hence, for the graph of given function, the axis of symmetry is $x=2.5$ . Hence, lightly sketch the line $x=2.5$ on the graph.

Now, the vertex of parabola is at $x=2.5$ . At $x=2.5$ , the $y-\text{coordinate}$ is

  $\begin{array}{l}$ y=-{(2.5)}^2+5(2.5)-3$\Rightarrow y=-6.25+12.5-3\\ \Rightarrow y=3.25\end{array}$

Hence, plot the vertex $(2.5,3.25)$ on the graph. Since $c=-3,$ the $y-\text{intercept}$ is $(0,-3).$ The reflection of $(0,-3)$ across $x=2.5$ is $(5,-3).$ Hence, plot both the points on the graph.

Since $a<0,$ the parabola opens downward. Sketching a smooth curve through the points to draw the graph.

Graph:

  

Interpretation:

From the graph, the minimum value is $y=-\infty$ and the maximum value is $y=3.25$ , so the range of the function is $(-\infty ,3.25]$ . Here, the function is defined for all values of $x.$ Hence, the domain for the given function is $(-\infty ,\infty ).$