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Chapter 1: Problem 72

Find the x- and y-intercepts of the graph of the equation. (a) \(4 x^{2}+25 y^{2}=100\) (b) \(x^{2}-x y+3 y=1\)

Short Answer

Expert verified
The x- and y-intercepts for (a) are (5,0), (-5,0) and (0,2), (0,-2). For (b), they are (1,0), (-1,0) and (0,1/3).

Step by step solution

01

Understand the Equation (a)

The given equation for part (a) is a standard form of an ellipse: \(4x^2 + 25y^2 = 100\). We need to find both x-intercepts and y-intercepts.
02

Find x-Intercepts for (a)

To find the x-intercepts, set \(y = 0\) in the equation \(4x^2 + 25y^2 = 100\). This gives:\[ 4x^2 + 25(0)^2 = 100 \] \[ 4x^2 = 100 \] \[ x^2 = 25 \] \[ x = \pm 5 \]Thus, the x-intercepts are \((5, 0)\) and \((-5, 0)\).
03

Find y-Intercepts for (a)

To find the y-intercepts, set \(x = 0\) in the equation \(4x^2 + 25y^2 = 100\). This gives:\[ 4(0)^2 + 25y^2 = 100 \] \[ 25y^2 = 100 \] \[ y^2 = 4 \] \[ y = \pm 2 \]Thus, the y-intercepts are \((0, 2)\) and \((0, -2)\).
04

Understand the Equation (b)

The equation is \(x^2 - xy + 3y = 1\). It is neither a standard form of ellipse or hyperbola, as it involves a mixed term \(xy\). We must find each intercept by substitution.
05

Find x-Intercepts for (b)

For x-intercepts, set \(y = 0\) in the equation \(x^2 - xy + 3y = 1\):\[ x^2 - x(0) + 3(0) = 1 \]\[ x^2 = 1 \]\[ x = \pm 1 \]Thus, the x-intercepts are \((1, 0)\) and \((-1, 0)\).
06

Find y-Intercepts for (b)

For y-intercepts, set \(x = 0\) in the equation \(x^2 - xy + 3y = 1\):\[ (0)^2 - 0y + 3y = 1 \]\[ 3y = 1 \]\[ y = \frac{1}{3} \]Thus, the single y-intercept is \((0, \frac{1}{3})\).

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Key Concepts

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

Ellipse
An ellipse is a curve on a plane that surrounds two focal points, such that the sum of the distances to the two foci from any point on the curve is constant. This is a special type of conic section formed by the intersection of a plane and a cone.

Ellipses have an oval shape and are often compared to circles, but they are elongated. In mathematics, the standard form of an ellipse can be written as:
  • \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \)
  • Where \(a\) and \(b\) represent the lengths of the semi-major and semi-minor axes respectively.
Ellipses often occur in physics and astronomy, like the orbits of planets, which are elliptical rather than perfectly circular.
Finding Intercepts
Intercepts are key in understanding the behavior of a graph. There are two primary intercepts in a graph: x-intercept and y-intercept.

  • **X-intercepts**: These are points where the graph crosses the x-axis. To find x-intercepts, set \(y = 0\) in the equation and solve for \(x\).
  • **Y-intercepts**: These occur where the graph crosses the y-axis. To find y-intercepts, set \(x = 0\) in the equation and solve for \(y\).
Finding intercepts can simplify the process of graphing equations, giving clear points where the line or curve will cross the axes. In ellipses or other conic sections, these points might indicate the starts of axes or reveal symmetry.
Precalculus
Precalculus serves as a foundational course that prepares students for calculus. It combines knowledge from algebra, trigonometry, and other mathematical principles to get students ready for the more advanced concepts that calculus introduces.

In precalculus, students learn about different mathematical functions and their properties. It includes the study of:
  • The behavior and analysis of functions, especially quadratic, polynomial, rational, exponential, and logarithmic functions.
  • Trigonometric functions and their applications.
  • Concepts of limits and introductory derivative concepts, which are cornerstones for calculus study.
Understanding how to manipulate and analyze various forms of equations, including finding intercepts and working with conic sections like ellipses, is a crucial skill developed in precalculus.

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Most popular questions from this chapter

Solving an Equation in Different Ways We have learned several different ways to solve an equation in this section. Some equations can be tackled by more than one method. For example, the equation \(x-\sqrt{x}-2=0\) is of quadratic type. We can solve it by letting \(\sqrt{x}=u\) and \(x=u^{2},\) and factoring. Or we could solve for \(\sqrt{x},\) square each side, and then solve the resulting quadratic equation. Solve the following equations using both methods indicated, and show that you get the same final answers. (a) \(x-\sqrt{x}-2=0 \quad\) quadratic type; solve for the radical, and square (b) \(\frac{12}{(x-3)^{2}}+\frac{10}{x-3}+1=0 \quad \begin{array}{l}\text { quadratic type; multiply } \\ \text { by } \mathrm{LCD}\end{array}\)

Use scientific notation, the Laws of Exponents, and a calculator to perform the indicated operations. State your answer rounded to the number of significant digits indicated by the given data. $$\frac{1.295643 \times 10^{9}}{\left(3.610 \times 10^{-17}\right)\left(2.511 \times 10^{6}\right)}$$

Falling Ball Using calculus, it can be shown that if a ball is thrown upward with an initial velocity of \(16 \mathrm{ft} / \mathrm{s}\) from the top of a building 128 ft high, then its height \(h\) above the ground \(t\) seconds later will be $$ h=128+16 t-16 t^{2} $$ During what time interval will the ball be at least \(32 \mathrm{ft}\) above the ground? (image cannot copy)

Simplify the expression. (a) \(\left(\frac{x^{3 / 2}}{y^{-1 / 2}}\right)^{4}\left(\frac{x^{-2}}{y^{3}}\right)\) (b) \(\left(\frac{4 y^{3} z^{2 / 3}}{x^{1 / 2}}\right)^{2}\left(\frac{x^{-3} y^{6}}{8 z^{4}}\right)^{1 / 3}\)

Distances in a City \(A\) city has streets that run north and south and avenues that run east and west, all equally spaced. Streets and avenues are numbered sequentially, as shown in the figure. The walking distance between points \(A\) and \(B\) is 7 blocks - that is, 3 blocks east and 4 blocks north. To find the straight-line distance \(d\), we must use the Distance Formula. (a) Find the straight-line distance (in blocks) between \(A\) and \(B\) (b) Find the walking distance and the straight-line distance between the corner of 4 th St. and 2 nd Ave. and the corner of 11 th St. and 26 th Ave. (c) What must be true about the points \(P\) and \(Q\) if the walking distance between \(P\) and \(Q\) equals the straight[line distance between \(P\) and \(Q ?\)
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