Answer to Question #239385 in Calculus for nickname2299

Question

1. Find a formula for the function whose graph consists of the line segment from the point (−4,3) to the point (−2,0) and the lower half of the circle centered at the origin with radius 2.

Answer 1

To determine:

An expression function for the graph which satisfies the given condition:

Given:

The graph has a line segment connecting (-4,3) and (-2,0) and it consists of a top half of the circle with center (0,0) and radius 2.

Calculation:

Find the slop of the line segment joining the points (-4,3) and (-2,0) as follows:

$m={y_2-y_1 \above{2pt} x_2-x_1}$

$m={0-3 \above{2pt} -2-(-4)}$

$m={-3 \above{2pt} 2}$

Thus, the slope of the line segment is $m=-\frac{3}{2}$ .

Find the y-intercept of the line segment joining the points (-4,3) and (-2,0) as follows

$y=mx+c$

$0=(-\frac{3}{2})(-2)+c$ 

$0=3+c$

$c=-3$

Thus , y intercept is c=-3.

The equation of the line segment is $y=-\frac{3}{2}x-3$ for $-4\leqslant x<-2$ .

Find the equation of the circle:

The equation of the circle of radius 2 centered on the origin is $x^2+y^2=4$

We can then solve for y

$x^2+y^2=4\iff$ $y^2=4-x^2\iff$ $y=\sqrt{4-x^2}$

To only include the top half of the circle we only take the positive root .

The equation of the circle is $\sqrt{x^2-4}$ for -22

$f(x)=\begin{cases} -\frac{3}{2}x-3 &\text{ on} [-4,-2) \\ \sqrt{4-x^2} &\text{on} [-2,2] \end{cases}$