Math Problem Statement

For the given parametric equations, find the points (x, y) corresponding to the parameter values t = -2, -1, 0, 1, 2. The parametric equations are: x = ln(5t^2 + 1), y = t / (t + 5).

Solution

Let's solve the problem using the given parametric equations:

x=ln(5t2+1),y=tt+5x = \ln(5t^2 + 1), \quad y = \frac{t}{t+5}

We need to find the points (x,y)(x, y) for t=0t = 0.

  1. For xx:

    x=ln(5(0)2+1)=ln(1)=0x = \ln(5(0)^2 + 1) = \ln(1) = 0

  2. For yy:

    y=00+5=0y = \frac{0}{0 + 5} = 0

Therefore, the point corresponding to t=0t = 0 is (x,y)=(0,0)(x, y) = (0, 0).

Would you like details on other values of tt or have any other questions?

Here are some related questions to consider:

  1. What are the points for t=2,1,1,2t = -2, -1, 1, 2?
  2. How do logarithmic functions behave as their argument approaches zero?
  3. What happens to the parametric equation y=tt+5y = \frac{t}{t+5} as tt approaches large positive or negative values?
  4. How can we plot the parametric curve for this range of tt?
  5. What would happen if we changed the parameters in the equations?

Tip: Remember, the natural logarithm function, ln(x)\ln(x), is only defined for positive values of xx.

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Math Problem Analysis

Mathematical Concepts

Parametric Equations
Logarithmic Functions
Rational Functions

Formulas

x = ln(5t^2 + 1)
y = t / (t + 5)

Theorems

Properties of logarithms
Behavior of rational functions

Suitable Grade Level

Grades 11-12