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Find an expression for the function whose graph consist of the line segment from the point $(1,-3)$ to the point $(5,7)$ together with the top of the circle with the center at $(8,7)$.

I don't understand what it means by together with a circle.

$$y = -\frac{5}{2}x + \frac{39}{2}$$

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To start with, note that the line segment has slope $(7-(-3))/(5-1)$, which is $10/4$. So your equation for the line segment part has some glitches. – André Nicolas Sep 8 '12 at 23:02
When you have a function with two pieces, as in your homework, you need to find equations for both of them. From there, you can use the notation for a piece-wise function to give your final answer. If you don't know what a piece-wise function is, you should look at the index in your textbook or google it. – Code-Guru Sep 8 '12 at 23:23
up vote 0 down vote accepted

I will assume that the circle has radius equal to $3$. If that is the case, then the circle is defined by

$C := \{ (x,y) \in \mathbb{R}^2 \mid (x-8)^2 + (y-7)^2 = 3^2\}$.

The equation that defines the circle can be rewritten in the form

$(y-7)^2 = 9 - (x-8)^2$

and, taking the square root of both sides, we obtain

$y = 7 \pm \sqrt{9 - (x-8)^2}$.

Since we want the top half of the circle, we have $y = 7 + \sqrt{9 - (x-8)^2}$.

The line that connects points $(1,-3)$ and $(5,7)$ is given by the equation

$y + 3 = \frac{10}{4} (x-1)$

or, equivalently,

$y = \displaystyle\frac{5}{2} x - \frac{11}{2}$.

Finally, we can introduce a function $f : [1,11] \to \mathbb{R}$ defined by

$f (x) = \left\{\begin{array}{cl} \displaystyle\frac{5}{2} x - \frac{11}{2}, &\quad{} x \in [1,5]\\ 7 + \sqrt{9 - (x-8)^2}, &\quad{} x \in [5,11]\end{array}\right.$

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It seems reasonable to assume that the graph of the function is as shown below:

$\hspace{5cm}$enter image description here

The red line is the line from the point $(1,-3)$ to $(5,7)$ and the green arc is the top of the circle with center $(8,7)$. The radius of the circle is implied by the distance to the point $(5,7)$ so that the graph is connected.

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Cool plot! How did you generate it? What software did you use? – Rod Carvalho Sep 8 '12 at 23:01
@RodCarvalho: I used Intaglio. – robjohn Sep 8 '12 at 23:05

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