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Magazine Chapter 9: Problem 34
Graph each parabola. Give the vertex, axis of symmetry, domain, and range. \(x=(y+3)^{2}-2\)
Short Answer
Expert verified
Vertex: (-2, -3), Axis: y = -3, Domain: \(\mathbb{R}\), Range: \([-2, \infty)\).
Step by step solution
01
Identify the vertex form of the equation
The given equation is \(x = (y+3)^2 - 2\). In this form, \((h, k)\) represents the vertex, where the equation is \(x = (y-k)^2 + h\). By comparison, we see that \(h = -2\) and \(k = -3\). Therefore, the vertex is \((-2, -3)\).
02
Determine the axis of symmetry
The axis of symmetry is a vertical line that passes through the vertex. Since the vertex is \((-2, -3)\), the axis of symmetry is \(y = -3\).
03
Identify the domain
For a parabola that opens left or right, the domain refers to the set of all possible \(x\) values. Given the equation \(x = (y+3)^2 - 2\), we see that it can take any real number as \((y+3)^2\) is always non-negative and can be extended to all real numbers. Hence, the domain is \(\mathbb{R}\) (all real numbers).
04
Identify the range
In this case, the parabola opens to the right (positive direction) since the coefficient of \((y+3)^2\) is positive. The minimum value of \(x\) is when \((y+3)^2 = 0\), which is \(x = -2\). Since the parabola opens to the right, the range of \(x\) is \([-2, \infty)\).
05
Graph the parabola
Plot the vertex at \((-2, -3)\). Sketch the parabola opening to the right from the vertex, as described. Draw the axis of symmetry as a vertical dashed line through \(y = -3\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
vertex of a parabola
Understanding the vertex of a parabola is crucial in graphing and analyzing parabolas. The vertex is the point where the parabola changes direction. For the given equation, \(x = (y + 3)^2 - 2\), we identify the vertex by comparing it to the standard form \(x = (y - k)^2 + h\). Here, \(h = -2\) and \(k = -3\). Therefore, the vertex is \((-2, -3)\). The vertex tells us the minimum or maximum point of the parabola, depending on its orientation.
axis of symmetry
The axis of symmetry is a vertical line that passes through the vertex of the parabola, dividing it into two mirror-image halves. For the equation \(x = (y + 3)^2 - 2\), the vertex is at \((-2, -3)\). Therefore, the axis of symmetry is the line \(y = -3\). This line helps in graphing as it ensures that each side of the parabola is symmetrical.
parabola domain
The domain of a parabola consists of all possible \(x\) values for which the parabola is defined. For the given parabola \(x = (y + 3)^2 - 2\), we see that \((y + 3)^2\) can take any real number value, which indicates the parabola can extend infinitely in both positive and negative directions along the \(y\)-axis. Thus, the domain of this parabola is \(\mathbb{R}\) (all real numbers), as \(x\) can be any real number.
parabola range
The range of a parabola is the set of all possible \(y\) values that the parabola can take. For the equation \(x = (y + 3)^2 - 2\), the parabola opens to the right, with the minimum \(x\)-value being \(-2\). This is because \((y + 3)^2\) is always non-negative, making \((y + 3)^2 - 2\) always greater than or equal to \(-2\). Therefore, the range of the given parabola is \([-2, \infty)\), meaning it includes all values starting from -2 and increasing to positive infinity.
graphing techniques
Graphing a parabola involves several steps: identifying the vertex, determining the axis of symmetry, and understanding the domain and range. Let's apply this to \(x = (y + 3)^2 - 2\).
1. **Plot the Vertex:** The vertex is at \((-2, -3)\).
2. **Draw the Axis of Symmetry:** This will be the line \(y = -3\).
3. **Determine the Orientation and Plot Points:** Since the parabola opens to the right, plot points around the vertex to see the shape more clearly. For instance, if \(y = -2\), then \(x = (y + 3)^2 - 2 = 1\). Plotting such points helps in drawing the full shape.
4. **Sketch the Parabola:** Connect the points smoothly, showing the opening towards the right.
Using these techniques will help ensure the parabola is graphed accurately.
1. **Plot the Vertex:** The vertex is at \((-2, -3)\).
2. **Draw the Axis of Symmetry:** This will be the line \(y = -3\).
3. **Determine the Orientation and Plot Points:** Since the parabola opens to the right, plot points around the vertex to see the shape more clearly. For instance, if \(y = -2\), then \(x = (y + 3)^2 - 2 = 1\). Plotting such points helps in drawing the full shape.
4. **Sketch the Parabola:** Connect the points smoothly, showing the opening towards the right.
Using these techniques will help ensure the parabola is graphed accurately.
