Chapter 10.2, Problem 52AYU

In Problems 39-56, find the vertex, focus, and directrix of each parabola. Graph the equation by hand. Verify your graph using a graphing utility.

$x^2-4x=\text{ 2}y$

To determine

(a & b)

To find: Vertex, focus and directrix of the parabola given by, $x²-4x=2y$.

Answer to Problem 52AYU

Vertex: $(2,-2)$.

Focus: $\left(2,-3/2\right)$.

Directrix D: $y=-5/2$.

Explanation of Solution

Given:

$x²-4x=2y$

Formula used:

Equation	Vertex	Focus	Directrix	Description
${\left(x-h\right)}^2=4a\left(y-k\right)$	$(h,k)$	$\left(h,k+a\right)$	$y=k-a$	Parabola, axis of symmetry is parallel to $y\text{-axis}$, opens up.

Calculation:

The equation, $x²-4x=2y$.

${\left(x-2\right)}^2=2\left(y+2\right)$ is of the form ${\left(x-h\right)}^2=4a\left(y-k\right)$.

$h=2;k=-2$

Hence, $4a=2;a=1/2$.

Therefore, vertex: $\left(h,k\right) =\left(2,-2\right)$.

Focus: $\left(h,a+k\right)=\left(2,-3/2\right)$.

Directrix D: $y=k-a=-2-1/2; y=-5/2$.

Parabola, axis of symmetry is parallel to $y\text{-axis}$, and it opens up.

To determine

c.

To graph: The parabola for the equation $x²-4x=2y$ by hand and verify using a graphing tool.

Answer to Problem 52AYU

Explanation of Solution

Given:

$x²-4x=2y$

Calculation:

The graph of the parabola $x²-4x=2y$.

$x²-4x=2y$

By using the table above, we obtain few points $\left(0,0\right),\left(4,0\right),\left(3,-1.5\right)$ and $\left(1,-1.5\right)$. We plot the graph of $x²-4x=2y$ using these points and the graph opens up. The points are shown above in the graph. Using graphing utility we verify the graph.

The Vertex $V\left(2,-2\right)$, focus $F\left(2,-3/2\right)$ is shown in the graph above.

The directrix D: $y=-\frac{5}{2}$ is shown in the dotted line.