### Video Transcript

There are two parts to this
question. The first part says we are given a
constant 𝑐. 𝐴 is the point where the line 𝑦
equals two 𝑐 to the power of 𝑥 plus three crosses the 𝑦-axis. Find the coordinates of the point
𝐴.

Any point that crosses the 𝑦-axis
has an 𝑥-coordinate equal to zero. This means that we can substitute
𝑥 equals zero into the equation 𝑦 equals two 𝑐 to the power of 𝑥 plus three. This gives us 𝑦 equals two 𝑐 to
the power of zero plus three. Zero plus three is equal to
three. Therefore, our 𝑦-coordinate is two
𝑐 cubed. The coordinates of the point 𝐴
where the line crosses the 𝑦-axis at zero, two 𝑐 cubed.

The second part of our question
says the following.

A circle has the equation 𝑥
squared plus 𝑦 squared minus 49 equals zero. The vector zero, two translates
circle 𝐶 to circle 𝐵. Sketch circle 𝐵. Clearly label the coordinates of
the center of circle 𝐵 and any points where 𝐵 intersects the 𝑦-axis.

The equation 𝑥 squared plus 𝑦
squared minus 49 equals zero can be rewritten as 𝑥 squared plus 𝑦 squared equals
49. This is the equation of a circle
with center zero, zero and radius root 49 which is equal to seven. The circle 𝐶, as shown in the
diagram, has center zero, zero. It intersects the 𝑥-axis at seven,
zero and minus seven, zero and intersects the 𝑦-axis at zero, seven and zero, minus
seven.

A translation of vector zero, two
will increase all the 𝑦-coordinates by two and will move the circle up the
coordinate grid. The center of the circle will move
from zero, zero to zero, two and the radius will remain equal to seven. Circle 𝐵 will, therefore,
intersect the 𝑦-axis at zero, nine and zero, minus five with a center of zero,
two.