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Magazine Chapter 12: Problem 72
Sketch the graph of each parabola. $$ x=(y+3)^{2}-1 $$
Short Answer
Expert verified
Vertex at \((-1, -3)\); Parabola opens to the right.
Step by step solution
01
- Identify the Equation Form
The provided equation is \(x = (y+3)^2 - 1\). This is a horizontal parabola in the form of \(x = a(y - k)^2 + h\), where the vertex form of the parabola gives us information about the vertex.
02
- Find the Vertex
From the equation \(x = (y + 3)^2 - 1\), we can see that the vertex \((h,k)\) is at \((h, k) = (-1, -3)\). Therefore, the vertex of this parabola is at \((-1, -3)\).
03
- Determine the Direction of Opening
Since the coefficient of \((y + 3)^2\) is positive, the parabola opens to the right.
04
- Calculate Additional Points
We need to determine a few points on either side of the vertex to sketch the graph effectively. Substitute some values of \(y\) (both greater than and less than -3) into the equation to get the corresponding x-coordinates: - For \(y = -2\), \( x = (-2 + 3)^2 - 1 = 1 - 1 = 0\). So, the point is \((0, -2)\). - For \(y = -4\), \( x = (-4 + 3)^2 - 1 = 1 - 1 = 0\). So, the point is \((0, -4)\). - For \(y = -1\), \( x = (-1 + 3)^2 - 1 = 4 - 1 = 3\). So, the point is \((3, -1)\). - For \(y = -5\), \( x = (-5 + 3)^2 - 1 = 4 - 1 = 3\). So, the point is \((3, -5)\).
05
- Sketch the Graph
Plot the vertex at \((-1, -3)\) and the points \((0, -2)\), \((0, -4)\), \((3, -1)\), and \((3, -5)\) on a coordinate plane. Draw a smooth curve through these points to reflect the parabola opening to the right.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Vertex Form of a Parabola
The vertex form of a parabola is a way to write the equation so that the vertex (the highest or lowest point) is easily identifiable. A general parabola equation in vertex form is \[x = a(y-k)^2 + h\], where \(h, k\) represents the vertex.
In the equation \[x = (y+3)^2 - 1\], you can see that it's written in vertex form. Here, \(h=-1\) and \(k=-3\). So, the vertex is at \((-1, -3)\).
To better understand this form, remember:
In the equation \[x = (y+3)^2 - 1\], you can see that it's written in vertex form. Here, \(h=-1\) and \(k=-3\). So, the vertex is at \((-1, -3)\).
To better understand this form, remember:
- \(h\) and \(k\): The vertex's coordinates.
- \(a\): Determines how wide or narrow the parabola is.
- Depending on whether the equation is \(x\) in terms of \(y\) or vice versa, it tells the direction of the parabola.
Graphing Parabolas
Graphing a parabola correctly involves several steps. For the equation \[x = (y+3)^2 - 1\], follow these steps:
1. **Find the Vertex**: As discussed earlier, the vertex is at \((-1, -3)\).
2. **Determine the Direction**: Since the coefficient of \((y + 3)^2\) is positive, the parabola opens to the right.
3. **Calculate Key Points**: Substitute \(y\) values around the vertex to find corresponding \(x\) coordinates:
4. **Plot Points on Coordinate Plane**: Mark the vertex and these key calculated points on the graph.
5. **Draw the Parabola**: Finally, draw a smooth curve passing through these points. Doing so helps to visually understand the shape and direction of the parabola.
1. **Find the Vertex**: As discussed earlier, the vertex is at \((-1, -3)\).
2. **Determine the Direction**: Since the coefficient of \((y + 3)^2\) is positive, the parabola opens to the right.
3. **Calculate Key Points**: Substitute \(y\) values around the vertex to find corresponding \(x\) coordinates:
- For \(y = -2\), \( x = 0\).
- For \(y = -4\), \( x = 0\).
- For \(y = -1\), \( x = 3\).
- For \(y = -5\), \( x = 3\).
4. **Plot Points on Coordinate Plane**: Mark the vertex and these key calculated points on the graph.
5. **Draw the Parabola**: Finally, draw a smooth curve passing through these points. Doing so helps to visually understand the shape and direction of the parabola.
Parabola Direction
The direction of a parabola indicates where it opens. For the given equation \[x = (y+3)^2 - 1\], since the \((y + 3)^2\) term is squared and the coefficient is positive, the parabola opens to the right.
If a coefficient is negative, the parabola opens in the opposite direction. Here鈥檚 an understanding of different directions based on the standard forms:
If a coefficient is negative, the parabola opens in the opposite direction. Here鈥檚 an understanding of different directions based on the standard forms:
- When the equation is in the form \(x = a(y - k)^2 + h\) with \(a > 0\), the parabola opens to the right.
- If \(a < 0\), it opens to the left.
- When the equation is \(y = a(x - h)^2 + k\) and \( a > 0\), it opens upward.
- If \(a < 0\), the parabola opens downward.
