Question

Find the center and radius of the circle, and sketch its graph.

$x^2+{\left(y-4\right)}^2=1$

Short answer

The center of the circle is $\left(0,4\right)$and radius is 1 unit. The graph of the circle is provided below.

Step-by-step solution

Step 1. Apply the concept of circles.

The standard form of the equation of the circle is ${\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2$, where$\left(h,k\right)$ denote the center of the circle and r denote the radius.

Step 2. Interpret the values of center and radius.

Consider the provided equation.

$x^2+{\left(y-4\right)}^2=1$

Rewrite the equation as provided below.

${\left(x-0\right)}^2+{\left(y-4\right)}^2=1^2$

Recall that the standard form of the equation of the circle is ${\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2$, where $\left(h,k\right)$ denote the center of the circle and r denote the radius.

Compare, $\left(h,k\right)$and $\left(0,4\right)$ also radius r is 1.

Here, $h=0,k=4\text{and}r=1$.

Therefore, the center of the circle is $\left(0,4\right)$and the radius is 1 unit.

Step 3. Graph the circle.

Now, on the Cartesian plane construct a circle with a center at $\left(0,4\right)$and radius of 1 unit.

The graph obtained is provided below,