Chapter 10.4, Problem 69AYU

To determine

To find: Analyze the equation.

Answer to Problem 69AYU

The equation represents parabola.

Explanation of Solution

Given:

$x²=16\left(y-3\right)$

Formula used:

Vertex	Focus	Directrix	Equation	Description
$\left(h,k\right)$	$\left(0,k+a\right)$	$y=k-a$	$\left(x-h\right)²=4a\left(y-k\right)$	Parabola, axis of symmetry is parallel to $y\text{-axis}$, opens up

Calculation:

Given that $x²=16\left(y-3\right)$.

The equation represents parabola.

The axis of symmetry is parallel to $y\text{-axis}$.

Vertex: $V\left(h,k\right)=\left(0,3\right)$.

Hence, $h=0,k=3$.

$4a=16$, $a=4$

Focus: $F\left(h,k+a\right)=F\left(0,7\right)=F\left(0,7\right)$.

The coordinates of the focus is $F\left(0,7\right)$.

Directrix: $y=3-4=-1$.

The graph of the equation of $x²=16\left(y-3\right)$ is plotted.

Using the given equation $x²=16\left(y-3\right)$, we plot the graph using graphing tool.

The graph represents a parabola, with vertex $\left(0,3\right)$, directrix $y=-1$ and the axis of symmetry is $y\text{-axis}$, opens up.