Questions: Find two different parametric representations for the equation of the parabola. y=(x+5)^2-5 (a) Let one parametric equation be x=t. Find the parametric equation for y. y= for t in (-∞, ∞) (Do net-factor.)

Find two different parametric representations for the equation of the parabola.
y=(x+5)^2-5
(a) Let one parametric equation be x=t. Find the parametric equation for y.
y= for t in (-∞, ∞)
(Do net-factor.)

Transcript text: Find two different parametric representations for the equation of the parabola. \[ y=(x+5)^{2}-5 \] (a) Let one parametric equation be $\mathrm{x}=\mathrm{t}$. Find the parametric equation for y . \[ y=\square \text { for } t \text { in }(-\infty, \infty) \] (Do net-factor.)

Solution

Solution Steps

Step 1: Identify the Given Equation

The given equation of the parabola is: \[ y = (x + 5)^2 - 5 \]

Step 2: Parametric Representation with \( x = t \)

Let \( x = t \). Substitute \( x = t \) into the equation to find the parametric equation for \( y \): \[ y = (t + 5)^2 - 5 \]

Step 3: Simplify the Equation

Expand and simplify the equation for \( y \): \[ y = t^2 + 10t + 25 - 5 \] \[ y = t^2 + 10t + 20 \]

Final Answer

The parametric equations are: \[ x = t \] \[ y = t^2 + 10t + 20 \quad \text{for } t \text{ in } (-\infty, \infty) \] \[ \boxed{y = t^2 + 10t + 20} \]

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