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Chapter 2: Problem 15

11鈥�18 ? Find the x- and y-intercepts of the graph of the equation. $$ x^{2}+y^{2}=4 $$

Short Answer

Expert verified
X-intercepts: (2, 0), (-2, 0); Y-intercepts: (0, 2), (0, -2).

Step by step solution

01

Understand Intercepts

Intercepts are points where the graph intersects the axes. The x-intercept is where the graph crosses the x-axis, meaning y = 0, while the y-intercept is where the graph crosses the y-axis, meaning x = 0.
02

Finding the x-intercepts

To find x-intercepts, set y = 0 and solve for x in the equation. Start with the equation: \(x^2 + y^2 = 4\). Substitute \(y = 0\): \(x^2 + 0^2 = 4\), which simplifies to \(x^2 = 4\). Then solve for x: \(x = \pm 2\). Thus, the x-intercepts are (2, 0) and (-2, 0).
03

Finding the y-intercepts

To find y-intercepts, set x = 0 and solve for y in the equation. Start with the equation: \(x^2 + y^2 = 4\). Substitute \(x = 0\): \(0^2 + y^2 = 4\), which simplifies to \(y^2 = 4\). Then solve for y: \(y = \pm 2\). Thus, the y-intercepts are (0, 2) and (0, -2).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

equation of a circle
The equation of a circle in the standard form is expressed as \(x^2 + y^2 = r^2\), where \(r\) represents the radius of the circle, and the center is located at the origin, (0,0). This general form allows us to quickly identify key properties of the circle.
  • Radius: The radius \(r\) is the square root of the constant on the right side of the equation. For example, in \(x^2 + y^2 = 4\), the radius is \(\sqrt{4} = 2\).
  • Center: The center of this circle is at (0,0) since there is no \(h\) or \(k\) term.

The simplicity of this equation makes it easy to visualize and plot the circle. Each point \((x, y)\) that satisfies this equation is a point on the circumference of the circle.
finding intercepts
Finding intercepts involves determining where the figure crosses the axes. These are special points on the graph:
  • x-intercepts: Located where the graph touches or crosses the x-axis. To find them, set \(y = 0\) in the equation and solve for \(x\).
  • y-intercepts: Located where the graph touches or crosses the y-axis. To find these, set \(x = 0\) in the equation and solve for \(y\).

These intercepts are particularly useful as they provide clear and simple starting points to graph the circle. In our example with the circle equation \(x^2 + y^2 = 4\):
  • For the x-intercepts, substituting \(y = 0\) into the equation gives the solutions \(x = \pm 2\), producing intercept points (2, 0) and (-2, 0).
  • Similarly, for the y-intercepts, substituting \(x = 0\) gives \(y = \pm 2\), producing intercept points (0, 2) and (0, -2).
solving quadratic equations
Solving quadratic equations is a fundamental skill in mathematics, encountered frequently when identifying intercepts for circles and other curves. A quadratic equation typically takes the form \(ax^2 + bx + c = 0\). Here's how to go about solving it:
  • Factoring: If the equation can be factored easily, solve for roots by setting each factor equal to zero.
  • Square Root Method: Used when the equation is in the form \(x^2 = d\). Directly take the square root on both sides to solve for \(x\).
  • Quadratic Formula: Apply when the equation does not easily factor: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

For the equation \(x^2 + 0 = 4\), the simplest method is the square root method. This yields \(x = \pm 2\), providing the solutions for x-intercepts directly and fast.

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Most popular questions from this chapter

Kepler鈥檚 Third Law Kepler鈥檚 Third Law of planetary motion states that the square of the period T of a planet (the time it takes for the planet to make a complete revolution about the sun) is directly proportional to the cube of its average distance d from the sun. (a) Express Kepler鈥檚 Third Law as an equation. (b) Find the constant of proportionality by using the fact that for our planet the period is about 365 days and the average distance is about 93 million miles. (c) The planet Neptune is about \(2.79 \times 10^{9} \mathrm{mi}\) from the sun. Find the period of Neptune.

Use a graphing device to graph the given family of lines in the same viewing rectangle. What do the lines have in common? \(y=2+m(x+3) \quad\) for \(m=0, \pm 0.5, \pm 1, \pm 2, \pm 6\)

Is Proportionality Everything? A great many laws of physics and chemistry are expressible as proportionalities. Give at least one example of a function that occurs in the sciences that is not a proportionality.

Growing Cabbages In the short growing season of the Canadian arctic territory of Nunavut, some gardeners find it possible to grow gigantic cabbages in the midnight sun. Assume that the final size of a cabbage is proportional to the amount of nutrients it receives, and inversely proportional to the number of other cabbages surrounding it. A cabbage that received 20 oz of nutrients and had 12 other cabbages around it grew to 30 lb. What size would it grow to if it received 10 oz of nutrients and had only 5 cabbage 鈥渘eighbors鈥�?

Use a graphing device to graph the given family of lines in the same viewing rectangle. What do the lines have in common? \(y=m(x-3) \quad\) for \(m=0, \pm 0.25, \pm 0.75, \pm 1.5\)
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