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Chapter 1: Problem 81

Test the equation for symmetry. $$y=x^{3}+10 x$$

Short Answer

Expert verified
The equation is symmetric about the origin only.

Step by step solution

01

Test for Symmetry about the x-axis

To test for symmetry about the x-axis, replace \( y \) with \( -y \) in the equation. The equation becomes: \(-y = x^3 + 10x\). Solve for \( y \): \( y = -x^3 - 10x\). The equation \( -y = x^3 + 10x \) does not match the original \( y = x^3 + 10x \), so it is not symmetric about the x-axis.
02

Test for Symmetry about the y-axis

To test for symmetry about the y-axis, replace \( x \) with \( -x \) in the equation. The equation becomes: \( y = (-x)^3 + 10(-x) \), which simplifies to: \( y = -x^3 - 10x \). The resulting equation \( y = -x^3 - 10x \) is not the same as the original \( y = x^3 + 10x \), indicating it is not symmetric about the y-axis.
03

Test for Symmetry about the Origin

To test for symmetry about the origin, replace \( x \) with \( -x \) and \( y \) with \( -y \). The equation becomes: \(-y = (-x)^3 + 10(-x)\), which simplifies to: \(-y = -x^3 - 10x\). Solving for \( y \), we get: \( y = x^3 + 10x \). This matches the original equation, indicating it is symmetric about the origin.

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Key Concepts

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

Symmetry About the X-axis
Symmetry about the x-axis occurs when a graph looks the same on the upper and lower halves of the x-axis. To test for this type of symmetry, we replace every instance of the variable \( y \) in the equation with \( -y \). For example, if the equation changes from \( y = x^3 + 10x \) to \( -y = x^3 + 10x \), and further manipulation yields \( y = -x^3 - 10x \), then:
  • The modified equation does not match the original equation.
  • This indicates that the graph is not symmetric about the x-axis.
When there is x-axis symmetry, points like \((x, y)\) will have mirrored points across the x-axis, like \((x, -y)\). If the equation doesn鈥檛 retain its original form after replacing \( y \) with \( -y \), then the curve is not symmetrical about the x-axis.
Symmetry About the Y-axis
Symmetry about the y-axis means that if you fold the graph along the y-axis, both sides match perfectly. To test this, substitute \( x \) with \( -x \) in the equation. For example, transforming \( y = x^3 + 10x \) to \( y = (-x)^3 + 10(-x) \) simplifies to \( y = -x^3 - 10x \).
  • The resulting equation is \( y = -x^3 - 10x \), which does not match the original equation.
  • This confirms that the graph is not symmetric about the y-axis.
If an equation is symmetric about the y-axis, the graph of points \((x, y)\) will reflect to \((-x, y)\). A mismatched result after substituting means the graph lacks this symmetry.
Symmetry About the Origin
Symmetry about the origin implies a kind of rotational symmetry, where if you rotate the graph 180 degrees, it looks the same. To test for this symmetry, replace both \( x \) with \( -x \) and \( y \) with \( -y \).For instance, starting with \( y = x^3 + 10x \), replace to get \(-y = (-x)^3 + 10(-x)\) which simplifies back to \( y = x^3 + 10x \).
  • The equation returns to its original form after replacing variables.
  • This indicates the graph is symmetric about the origin.
With origin symmetry, points \((x, y)\) reflect to \((-x, -y)\) across the origin. When the transformed equation equals the original, the graph has origin symmetry.

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Most popular questions from this chapter

(a) Show that \(a b=\frac{1}{2}\left[(a+b)^{2}-\left(a^{2}+b^{2}\right)\right]\) (b) Show that \(\left(a^{2}+b^{2}\right)^{2}-\left(a^{2}-b^{2}\right)^{2}=4 a^{2} b^{2}\) (c) Show that \(\left(a^{2}+b^{2}\right)\left(c^{2}+d^{2}\right)=(a c+b d)^{2}+(a d-b c)^{2}\) (d) Factor completely: \(4 a^{2} c^{2}-\left(a^{2}-b^{2}+c^{2}\right)^{2}\)

Use a Special Factoring Formula to factor the expression. $$1+1000 y^{3}$$

Factor the expression completely. $$\left(a^{2}+2 a\right)^{2}-2\left(a^{2}+2 a\right)-3$$

Factor the expression completely. $$t^{2}-6 t+9$$

Perform the indicated operations and simplify. $$(x+2)\left(x^{2}+2 x+3\right)$$
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