Question

2.3 homework question 6, 2.3.25 hw score: 23.81%, 5 of 21 points points: 0 of 1 sketch the graph of the first function. then, on the same coordinate plane, use a transformation to sketch the second graph. y = \sqrt{49 - x^{2}}, y = -\sqrt{49 - x^{2}} which graph shows y = \sqrt{49 - x^{2}} in red and y = -\sqrt{49 - x^{2}} in blue? a. b. c. d.

Explanation:

Step1: Analyze the first - function

The function $y = \sqrt{49 - x^{2}}$ can be rewritten as $y^{2}=49 - x^{2}$ or $x^{2}+y^{2}=49(y\geq0)$. This is the upper - half of a circle centered at the origin $(0,0)$ with radius $r = 7$ since for real - valued $y$, $49−x^{2}\geq0$, i.e., $- 7\leq x\leq7$.

Step2: Analyze the second - function

The function $y=-\sqrt{49 - x^{2}}$ can be rewritten as $y^{2}=49 - x^{2}$ or $x^{2}+y^{2}=49(y\leq0)$. This is the lower - half of a circle centered at the origin $(0,0)$ with radius $r = 7$.

Step3: Determine the correct graph

The graph of $y = \sqrt{49 - x^{2}}$ is the upper - half of the circle and $y=-\sqrt{49 - x^{2}}$ is the lower - half of the circle.

Answer:

The graph that shows the upper - half of the circle $x^{2}+y^{2}=49$ in red and the lower - half of the circle $x^{2}+y^{2}=49$ in blue is the correct one. Without seeing the actual options, the general shape should be a full circle centered at the origin with radius 7, where the upper - part is red and the lower - part is blue. If we assume the standard orientation of the coordinate axes, the correct graph is the one where the red curve is above the $x$ - axis and the blue curve is below the $x$ - axis for $x\in[-7,7]$.