91Ó°ÊÓ

To convert the polar equation into rectangular coordinates, we use the following formulas: $$ x = r \cos \theta\\ y = r \sin \theta $$ Given the polar equation: $$r = 4 \cos \theta + 4 \sin \theta$$ We can rewrite this as: $$r^2 = r(4\cos \theta + 4\sin \theta)$$ Replace \(r^2\) with \(x^2+y^2\), \(r\cos \theta\) with \(x\), and \(r\sin \theta\) with \(y\): $$ x^2 + y^2 = 4x + 4y $$

We will find the points where the equation intersects with the axes (when either x or y is zero): 1. Intersection with the x-axis (\((x,0)\)): Set y = 0 in the equation: $$ x^2 + 0^2 = 4x + 4(0) $$ $$ x^2 - 4x = 0 $$ Factor out x: $$ x(x-4) = 0 $$ So the points of intersection with the x-axis are (0,0) and (4,0). 2. Intersection with the y-axis (\((0, y)\)): Set x = 0 in the equation: $$ 0^2 + y^2 = 4(0) + 4y $$ $$ y^2 - 4y = 0 $$ Factor out y: $$ y(y-4) = 0 $$ So the points of intersection with the y-axis are (0,0) and (0,4).

Based on the rectangular equation (\(x^2+y^2=4x+4y\)) and the intersection points, the curve will be symmetric with respect to the x-axis and the y-axis. 1. Start by plotting the intersection points: (0,0), (4,0), (0,4). 2. Observe that the equation can be rewritten as: $$x^2 - 4x + y^2 - 4y = 0$$ 3. We can add \(4^2/2^2=4\) to both sides of the equation to complete the squares for both x and y: $$x^2 - 4x + 4 + y^2 - 4y + 4 = 8$$ 4. Factoring the completed squares gives: $$(x - 2)^2 + (y - 2)^2 = 8$$ This equation represents a circle with a center at (2,2) and a radius of \(\sqrt{8}\). This gives us a clear idea of the shape of the curve. 5. Now, we can sketch the circle using the information above. Draw the circle centered at (2,2) with radius \(\sqrt{8}\), which passes through the intersection points (0,0), (4,0), and (0,4). Thus, we have sketched the graph of the polar equation $$r=4 \cos \theta+4 \sin \theta$$.