Math Problem Statement

Graph the plane curve given by the parametric equations: x = 4t - 1, y = t^2, -4 ≤ t ≤ 4. Then find an equivalent rectangular equation and choose the correct graph.

Solution

Let’s solve this step by step to determine the correct graph for the given parametric equations:

The parametric equations are:
x=4t1x = 4t - 1,
y=t2y = t^2,
with the parameter 4t4-4 \leq t \leq 4.

Step 1: Eliminate tt to find a rectangular equation.

From x=4t1x = 4t - 1, solve for tt: t=x+14.t = \frac{x + 1}{4}.

Substitute this expression for tt into y=t2y = t^2: y=(x+14)2.y = \left(\frac{x + 1}{4}\right)^2.

Simplify: y=(x+1)216.y = \frac{(x + 1)^2}{16}.

So the rectangular equation is: y=(x+1)216.y = \frac{(x + 1)^2}{16}.

Step 2: Determine the range of xx and yy.

  • From x=4t1x = 4t - 1, when t=4t = -4:
    x=4(4)1=17x = 4(-4) - 1 = -17.
    When t=4t = 4:
    x=4(4)1=15x = 4(4) - 1 = 15.
    So, 17x15-17 \leq x \leq 15.

  • From y=t2y = t^2, since t20t^2 \geq 0 for all tt:
    y0y \geq 0. The maximum value of yy occurs when t=±4t = \pm 4, giving y=16y = 16.
    So, 0y160 \leq y \leq 16.

Step 3: Identify the correct graph.

The graph is a parabola opening upwards, with its vertex at (x,y)=(1,0)(x, y) = (-1, 0) (since x=1x = -1 when t=0t = 0).

  • The parabola is symmetric about x=1x = -1.
  • The yy-values range from 00 to 1616.

From the options:

  • Option C is correct.

Would you like further details on the solution or a graph sketch?

Questions to consider:

  1. How would the graph change if x=4t+cx = 4t + c instead of x=4t1x = 4t - 1?
  2. What happens if the parameter range is extended beyond 4t4-4 \leq t \leq 4?
  3. Can you identify the effect of coefficients in x=4t1x = 4t - 1 and y=t2y = t^2 on the parabola's shape?
  4. How would you check the symmetry of a parametric curve analytically?
  5. What other methods exist to convert parametric equations to rectangular form?

Tip:

When eliminating parameters, always double-check for restrictions on tt to ensure the graph corresponds to the correct portion of the curve.

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Math Problem Analysis

Mathematical Concepts

Parametric Equations
Elimination of Parameters
Quadratic Functions
Graphing

Formulas

t = (x + 1) / 4
y = ((x + 1)^2) / 16

Theorems

Vertex Form of a Parabola

Suitable Grade Level

Grades 10-12