Chapter 8.1, Problem 31E

Solution

To sketch: the graph of the parabola $y^2=-4x$ by hand.

Given:

The equation of a parabola: $y^2=-4x$ .

Graph:

The general form of the equation of a parabola that has the vertex $\left(0,0\right)$ and focus $\left(p,0\right)$ is $y^2=4px$ .

Observe that the equation $y^2=-4x$ is in the form $y^2=4px$ where $p=-1$ .

Thus, $y^2=-4x$ is the equation of the parabola having the vertex $\left(0,0\right)$ and focus $\left(-1,0\right)$ .

Plot the vertex $\left(0,0\right)$ and the focus $\left(-1,0\right)$ of the parabola on a coordinate grid as shown below:

  

Since the parabola has the vertex $\left(0,0\right)$ and the focus $\left(-1,0\right)$ , it is opened to the left side of the $x$ -axis.

That is, the possible $x$ values of the parabola $y^2=-4x$ are all negative.

Now, find few points on the parabola for the purpose of drawing it accurately.

For that purpose, substitute $x=-1$ in the equation $y^2=-4x$ and solve it for $y$ :

  $\begin{array}{c}{y}^2=-4\cdot \left(-1\right)\\ y^2=4\\ y=\pm \sqrt{4}\\ y=-2,2\end{array}$ .

Thus, the points $\left(-1,-2\right)$ and $\left(-1,2\right)$ lie on the parabola $y^2=-4x$ .

Similarly, it may be observed that the points $\left(-4,4\right)$ and $\left(-4,-4\right)$ lie on the parabola $y^2=-4x$ .

Plot the points $\left(-1,-2\right)$ , $\left(-1,2\right)$ , $\left(-4,-4\right)$ and $\left(-4,4\right)$ along with the vertex and the focus of the parabola on the same grid:

  

Then, join the points plotted previously and the vertex of the parabola using a smooth curve to obtain the graph of the parabola as shown below: