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Magazine Chapter 8: Problem 115
Graph each parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range. $$y^{2}-4 y+4=4 x+4$$
Short Answer
Expert verified
Vertex: (1, 2); Axis: y = 2; Domain: [1, 鈭�); Range: (-鈭�, 鈭�).
Step by step solution
01
Rewrite the Equation
First, let's rewrite the equation in a more recognizable form for graphing. The given equation is \(y^2 - 4y + 4 = 4x + 4\). We rearrange this as \((y^2 - 4y + 4) - 4 = 4x\), which simplifies to \((y - 2)^2 = 4(x - 1)\). This is the standard form of a parabola \((y-k)^2 = 4p(x-h)\) that opens sideways.
02
Identify the Vertex
In the equation \((y - 2)^2 = 4(x - 1)\), the vertex is given by \((h, k)\). Here, \(h = 1\) and \(k = 2\). Therefore, the vertex of the parabola is \((1, 2)\).
03
Determine the Axis of Symmetry
The axis of symmetry for a horizontal parabola \((y - k)^2 = 4p(x - h)\) is the horizontal line passing through the vertex, parallel to the y-axis. Therefore, the equation of the axis of symmetry is \(y = 2\).
04
Calculate the Focus and Directrix
Here, \(4p = 4\) gives \(p = 1\). Since the parabola opens to the right, the focus is \((h+p, k) = (1+1, 2) = (2, 2)\). The directrix will be \(x = h - p = 1 - 1 = 0\).
05
Determine the Domain and Range
Since the parabola opens to the right, the domain is \(x \geq 1\) or \([1, \infty)\). The range includes all real values of \(y\), i.e., \((-fty, \infty)\).
06
Sketch the Parabola and Verify
Draw the parabola with vertex at \((1, 2)\), extending to the right. Use the focus \((2, 2)\) and the directrix \(x = 0\) to guide the accurate shape. Verify by entering \((y-2)^2 = 4(x-1)\) into a graphing calculator and check if it matches the sketch.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Vertex Form
Understanding the vertex form of a parabola is crucial for graphing it easily. In the equation \[(y - k)^2 = 4p(x - h)\]or alternatively, if a parabola opens up or down \[(x - h)^2 = 4p(y - k)\],\((h, k)\) are the coordinates of the vertex of the parabola. This vertex is the point where the parabola makes a turn. In our example, by converting the equation of the parabola to its vertex form, \((y - 2)^2 = 4(x - 1)\),you can quickly determine the vertex to be \((1, 2)\).This means the parabola opens horizontally and the vertex is a key point on the graph.
Axis of Symmetry
The axis of symmetry in a parabola is an imaginary line that divides it into two mirror-image halves. For a sideways-opening parabola like \[(y - k)^2 = 4p(x - h)\],the axis of symmetry is parallel to the y-axis. This axis is a vertical line that passes through the vertex of the parabola. It can be represented by the equation \(y = k\), where \(k\) is the y-coordinate of the vertex. In our parabola \((y - 2)^2 = 4(x - 1)\),the axis of symmetry is \(y = 2\), indicating the parabola is split symmetrically around this line.
Domain and Range
The domain and range of a parabola tell us which x-values and y-values, respectively, the graph includes. For our sideways parabola, \((y - 2)^2 = 4(x - 1)\),the opening direction influences these values. When a parabola opens right or left, the domain is restricted. Here, the domain is \(x \geq 1\), or \([1, \infty)\), because the graph starts at x = 1 and extends indefinitely to the right. Conversely, the range of the parabola is unrestricted, covering all real numbers or \((-\infty, \infty)\),since the y-values can extend infinitely vertically in both directions.
Focus and Directrix
The focus and directrix of a parabola are essential for understanding its shape and direction. The focus is a point from which distances are measured to form the set of points on the curve. The directrix is a line that does the same, acting as a constant reference. For the equation \((y - 2)^2 = 4(x - 1)\),since \(4p = 4\),we find \(p = 1\). This makes the focus \((h + p, k) = (2, 2)\). The focus lies along the axis of symmetry but at a distance of \(p\) from the vertex. On the other side of the vertex at the same distance \(-p\),the directrix is a vertical line, here \(x = 0\). These definitions show how the parabola's curve balances around equal distances from the focus and directrix, creating its symmetric shape.
