Questions: For the given equation, solve the following: x=-3(y+5)^2-8 Part 1 out of 3 Find the x - and y-intercepts (if they exist) and the vertex of the parabola. Enter the intercepts as a comma-separated list of points, if necessary. If an intercept does not exist, enter DNE in the appropriate field. The x-intercept(s) are The y-intercept(s) are The vertex is

For the given equation, solve the following: x=-3(y+5)^2-8 Part 1 out of 3 Find the x - and y-intercepts (if they exist) and the vertex of the parabola. Enter the intercepts as a comma-separated list of points, if necessary. If an intercept does not exist, enter DNE in the appropriate field. The x-intercept(s) are The y-intercept(s) are The vertex is
Transcript text: For the given equation, solve the following: \[ x=-3(y+5)^{2}-8 \] Part 1 out of 3 Find the $x$ - and $y$-intercepts (if they exist) and the vertex of the parabola. Enter the intercepts as a comma-separated list of points, if necessary. If an intercept does not exist, enter DNE in the appropriate field. The $x$-intercept(s) are $\square$ The $y$-intercept(s) are $\square$ The vertex is $\square$
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Solution

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Solution Steps

To find the intercepts and the vertex of the given parabola, we can follow these steps:

  1. Y-intercept: Set \( y = 0 \) in the equation and solve for \( x \).
  2. X-intercept: Set \( x = 0 \) in the equation and solve for \( y \). If the solution is not real, the x-intercept does not exist.
  3. Vertex: The vertex form of a parabola is \( x = a(y - k)^2 + h \). In this equation, the vertex is at \( (h, k) \). Rewrite the equation to identify \( h \) and \( k \).
Step 1: Finding the Y-Intercept

To find the \( y \)-intercept, we set \( y = 0 \) in the equation \( x = -3(y + 5)^2 - 8 \):

\[ x = -3(0 + 5)^2 - 8 = -3(25) - 8 = -75 - 8 = -83 \]

Thus, the \( y \)-intercept is \( (0, -83) \).

Step 2: Finding the X-Intercept

To find the \( x \)-intercept, we set \( x = 0 \) in the equation:

\[ 0 = -3(y + 5)^2 - 8 \]

Rearranging gives:

\[ 3(y + 5)^2 = -8 \]

Since the left side is non-negative and the right side is negative, there are no real solutions. Therefore, the \( x \)-intercepts do not exist.

Step 3: Finding the Vertex

The vertex of the parabola can be identified from the equation in vertex form. The equation is given as:

\[ x = -3(y + 5)^2 - 8 \]

From this, we can see that the vertex is at the point \( (-8, -5) \).

Final Answer

The \( y \)-intercept is \( (0, -83) \), the \( x \)-intercepts do not exist (DNE), and the vertex is \( (-8, -5) \).

\[ \boxed{\text{Y-Intercept: } (0, -83), \text{ X-Intercept: DNE, Vertex: } (-8, -5)} \]

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