Math Problem Statement
Solution
Let’s solve this step by step to determine the correct graph for the given parametric equations:
The parametric equations are:
,
,
with the parameter .
Step 1: Eliminate to find a rectangular equation.
From , solve for :
Substitute this expression for into :
Simplify:
So the rectangular equation is:
Step 2: Determine the range of and .
-
From , when :
.
When :
.
So, . -
From , since for all :
. The maximum value of occurs when , giving .
So, .
Step 3: Identify the correct graph.
The graph is a parabola opening upwards, with its vertex at (since when ).
- The parabola is symmetric about .
- The -values range from to .
From the options:
- Option C is correct.
Would you like further details on the solution or a graph sketch?
Questions to consider:
- How would the graph change if instead of ?
- What happens if the parameter range is extended beyond ?
- Can you identify the effect of coefficients in and on the parabola's shape?
- How would you check the symmetry of a parametric curve analytically?
- What other methods exist to convert parametric equations to rectangular form?
Tip:
When eliminating parameters, always double-check for restrictions on to ensure the graph corresponds to the correct portion of the curve.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
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
Related Recommendation
Graphing Parametric Equations: x = (1/4)t, y = 3t - 8
Parametric Equation Conversion: Sketching and Eliminating Parameters
Sketching Parametric Curves: Quadratic and Linear Functions
Eliminate the Parameter t for Parametric Equations: x(t) = 5t², y(t) = -10 + 4t
Graphing Parametric Equations x = 2t and y = t + 1 for -2 < t < 2