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Chapter 1: Problem 6

Determine all intercepts of the graph of the equation. Then decide whether the graph is symmetric with respect to the \(x\) axis, the \(y\) axis, or the origin. $$ x^{4}=3 y^{3}+4 $$

Short Answer

Expert verified
X-intercepts: \((\sqrt{2}, 0), (-\sqrt{2}, 0)\); Y-intercept: \((0, -\sqrt[3]{4/3})\). The graph is symmetric with respect to the y-axis.

Step by step solution

01

Determine the x-intercepts

To find the x-intercepts, set \( y = 0 \) in the equation and solve for \( x \). The equation becomes \( x^4 = 3(0)^3 + 4 \), which simplifies to \( x^4 = 4 \). Solving \( x^4 = 4 \) gives \( x = \pm \sqrt[4]{4} \), which are \( x = \pm \sqrt{2} \). Therefore, the x-intercepts are \( (\sqrt{2}, 0) \) and \( (-\sqrt{2}, 0) \).
02

Determine the y-intercepts

To find the y-intercepts, set \( x = 0 \) in the equation and solve for \( y \). The equation becomes \( 0^4 = 3y^3 + 4 \), which simplifies to \( 0 = 3y^3 + 4 \). Solving \( 0 = 3y^3 + 4 \) gives \( y^3 = -\frac{4}{3} \), so \( y = -\sqrt[3]{\frac{4}{3}} \). Thus, the y-intercept is \( (0, -\sqrt[3]{\frac{4}{3}}) \).
03

Test for symmetry with respect to the x-axis

Replace \( y \) with \( -y \) to test for symmetry with respect to the x-axis. The equation becomes \( x^4 = 3(-y)^3 + 4 \), which simplifies to \( x^4 = -3y^3 + 4 \). Since this is not equivalent to the original equation \( x^4 = 3y^3 + 4 \), the graph is not symmetric with respect to the x-axis.
04

Test for symmetry with respect to the y-axis

Replace \( x \) with \( -x \) to test for symmetry with respect to the y-axis. The equation becomes \( (-x)^4 = 3y^3 + 4 \), which simplifies to \( x^4 = 3y^3 + 4 \). Since this equation is equivalent to the original equation, the graph is symmetric with respect to the y-axis.
05

Test for symmetry with respect to the origin

Replace \( x \) with \( -x \) and \( y \) with \( -y \) to test for symmetry with respect to the origin. The equation becomes \( (-x)^4 = 3(-y)^3 + 4 \), which simplifies to \( x^4 = -3y^3 + 4 \). Since this is not equivalent to the original equation \( x^4 = 3y^3 + 4 \), the graph is not symmetric with respect to the origin.

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Key Concepts

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

Intercepts
Intercepts serve as the points at which a graph crosses the axes in a Cartesian coordinate system. Understanding intercepts is crucial because they help us visually interpret equations. There are two main types of intercepts: the x-intercept and the y-intercept. Each provides valuable insights into the behavior of an equation's graph. Let's dive deeper into what each type of intercept represents and how to calculate them.
X-Intercept
The x-intercepts of a graph are the points where the graph crosses the x-axis. To find the x-intercepts of an equation, substitute 0 for the variable y and solve for x. In the given equation, setting \( y = 0 \) transforms it into \( x^4 = 4 \). Solving this, we find that \( x = \pm \sqrt{2} \). Thus, the graph intersects the x-axis at the points \( (\sqrt{2}, 0) \) and \( (-\sqrt{2}, 0) \). These intercepts are essential in sketching the graph as they provide exact points of intersection with the x-axis.
Y-Intercept
The y-intercept of a graph is the point where it crosses the y-axis. To find the y-intercept, you set x to zero and solve for y. In our example, by substituting \( x = 0 \) into the equation, we simplify it to \( 0 = 3y^3 + 4 \). Solving this for y yields \( y = -\sqrt[3]{\frac{4}{3}} \). Therefore, the y-intercept occurs at \( (0, -\sqrt[3]{\frac{4}{3}}) \). This point helps us understand where the graph intersects the y-axis, which aids in constructing an accurate representation of the graph's behavior.
Symmetry Tests
Symmetry tests help us understand the visual balance of a graph concerning particular axes or the origin. We have three common types of symmetry checks: with respect to the x-axis, the y-axis, and the origin.
  • X-axis Symmetry: Replace y with -y in the equation. If the resulting equation is identical to the original, the graph is symmetric about the x-axis. For our equation, this is not the case.
  • Y-axis Symmetry: Replace x with -x in the equation. If the modified equation is the same as the original, the graph is symmetric with respect to the y-axis. Here, this condition is satisfied.
  • Origin Symmetry: Replace both x with -x and y with -y. If the equation remains unchanged, the graph is symmetric about the origin. The given equation does not fulfill this condition.
Understanding these symmetry properties simplifies graph sketching and helps in identifying unique graph characteristics.

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Most popular questions from this chapter

Suppose a ball of mass \(m\) is attached to a string of length \(L\) and is rotated in a vertical plane with enough velocity \(v\) so that the string remains taut (Figure \(1.78)\). Then the tension \(T\) in the string, which depends on the angle \(\theta\) that the string makes with the downward vertical, is given by $$ T=m\left(\frac{v^{2}}{L}+g \cos \theta\right) \text { for } 0 \leq \theta<2 \pi $$ where \(g\) is the (negative) acceleration due to gravity. a. From your intuition, at which point in the path of the ball would the tension be greatest, and at which would it be least? b. From (14), find the value of \(\theta\) at which \(T\) is greatest and the value at which \(T\) is least. Do these values agree with your intuition?

Determine at which points the graphs of the given pair of functions intersect. $$ f(x)=e^{3 x} \text { and } g(x)=3 e^{x} $$

Starting at noon, \(A\) flies 2400 miles from New York to San Francisco at a velocity of 400 miles per hour. \(B\) starts the same trip at \(2: 00\) P.M. the same day with a velocity of 800 miles per hour. Express the distance \(D\) between \(A\) and \(B\) at any instant between noon and \(5: 00 \mathrm{P} . \mathrm{M}\). in terms of the time in hours elapsed after noon.

Let \(a\) and \(b\) be distinct positive numbers different from \(1 .\) Show that \(\log _{a} x \neq \log _{b} x\) for \(x \neq 1\).

Let \(f\) be a function, and let \(g(x)=f(x)-3 .\) What is the relationship between the graphs of \(f\) and \(g\) ?
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