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Magazine Chapter 20: Problem 15
Sketch the curves depicting the following equations: (a) \(x=\sqrt{9-y^{2}}\) (b) \(y^{2}=16 x\) (c) \(x y=5\)
Short Answer
Expert verified
(a) Semi-circle, (b) Right-opening parabola, (c) Hyperbola.
Step by step solution
01
Understand the Equation (a)
The equation given is \(x=\sqrt{9-y^{2}}\). This is the equation of a semi-circle. Notice that solving for \(x\) gives us \(x^2 = 9 - y^2\), which can be rearranged as \(x^2 + y^2 = 9\). This is the equation of a circle centered at the origin with radius 3. Since \(x=\sqrt{9-y^{2}}\), we consider only the positive \(x\) values, implying the right half (positive \(x\)-axis) of the circle.
02
Sketch the Curve (a)
Draw a semi-circle on the positive side of the x-axis, centered at the origin \((0, 0)\) with a radius of 3. The curve will touch the point \((3, 0)\) on the x-axis because when \(y = 0\), \(x = 3\). Similarly, it will touch the points \((0, 3)\) and \((0, -3)\) on the y-axis.
03
Understand the Equation (b)
The equation \(y^{2}=16x\) can be identified as a parabola. Rewrite it as \(y = \pm \sqrt{16x}\) or \(y = \pm 4\sqrt{x}\). This indicates that the parabola opens to the right with the vertex at the origin \((0, 0)\). The parabola is symmetric with respect to the x-axis because of the \( \pm \) sign.
04
Sketch the Curve (b)
Draw a parabola opening to the right, starting from the origin. The arms should stretch positively and negatively along the y-axis as \(x\) increases. This results in the curve crossing the y-axis symmetrically, indicating the presence of \((y^2)\) in the original equation.
05
Understand the Equation (c)
The equation \(xy = 5\) is a rectangular hyperbola. Rearrange it to \(y = \frac{5}{x}\) to better understand the shape. This graph will approach the x-axis and y-axis asymptotically, meaning it never actually touches these axes.
06
Sketch the Curve (c)
Draw a hyperbola in two separate branches in the first and third quadrants as \(x\) cannot be zero. The curve approaches the x- and y-axes but does not intersect them, exhibiting an asymptotic behavior as \(x\rightarrow 0\) and \(x\rightarrow \infty\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Semi-circle Graph
A semi-circle graph emerges from partial representations of a complete circle. In equation form, it often presents as half of a perfectly formed circle. Consideration of the equation \(x = \sqrt{9 - y^2}\) shows this. Here, the full circle equation \(x^2 + y^2 = 9\) is split, giving us a semi-circle.
- The equation \(x = \sqrt{9 - y^2}\) implies only positive values for \(x\).
- This means we see the half of the circle on the right, centered at the origin \((0,0)\).
- The radius is 3, as seen from \(x^2 + y^2 = 9\), equivalent to the standard circle equation \(x^2 + y^2 = r^2\).
Parabola Equation
Parabolas are curves that form an open bow-like shape, typically defined by quadratic equations. The focus here is on parabolas that open horizontally, like equation \(y^2 = 16x\). Rewriting gives \(y = \pm 4\sqrt{x}\), showcasing the right opening parabola.
- Vertex at the origin, point \((0,0)\) acts as the center point.
- The equation's symmetry emerges from the \(\pm\) sign, signaling it stretches on both positive and negative y-axes.
- Each arm of the parabola diverges from the vertex as \(x\) increases.
Rectangular Hyperbola
A rectangular hyperbola is a unique graph depicted by products of variables equating to constants, as seen in the equation \(xy = 5\). Rearrangement provides \(y = \frac{5}{x}\) for clarity and shows the asymptotic characteristics of this curve.
- Two branches exist, one each in the first and third quadrants.
- The graph never meets the x- or y-axes, illustrating its distinct asymptotes.
- These curves approach the axes infinitely close yet never intersect.
