Chapter 2.3, Problem 44E

To determine

To describe: Identify the vertex, focus, directrix and axis of symmetry of the parabola.

Answer to Problem 44E

  $\begin{array}{l}\text{Therefore, Vertex:}(-3,-5);\\ \text{Focus:}\left(-3,\frac{-19}{4}\right);\\ \text{Directrix:}y=-5-\frac{1}{4}=\frac{-21}{4}\\ \text{Axis of symmetry:}x=-3\\ \text{Transformation of the graph, }y=\frac{1}{4}{x}^2\end{array}$

Explanation of Solution

Given information: The equation of the parabola is,

  $y={(x+3)}^2-5$

Formula Used: The general equation of parabola,

  $y=\frac{1}{4p}{(x-h)}^2+k$

Calculation: To identify the vertex, focus, directrix and axis of symmetry for,

  $\begin{array}{l}$ y={(x+3)}^2-5$\text{On comparing with general form}\\ $ y=\frac{1}{4p}{(x-h)}^2+k$\text{where Vertex:}(h,k)\text{ and Focus:(}h,k+p)\\ \text{and Directrix:}y=k-p\\ $ y={(x+3)}^2-5$\text{Then, Vertex:}(-3,-5);& \text{Focus:}\left(-3,\frac{-19}{4}\right);\\ \text{Directrix:}y=-5-\frac{1}{4}=\frac{-21}{4}\\ \text{Axis of symmetry:}x=-3\end{array}$

The transformation of the graph of the standard equation with $p=1\text{and vertex}(0,0).$

Therefore, the equation is: $y=\frac{1}{4}{x}^2$