/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 3 Graph \(y=(x-2)^{2}+1\).... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 3

Graph \(y=(x-2)^{2}+1\).

Short Answer

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Answer: The key features of the graph of the quadratic function \(y = (x-2)^2 + 1\) are a vertex at the point (2,1), axis of symmetry at \(x=2\), and the parabola opens upwards.

Step by step solution

01

Determine the vertex

The given function is already in vertex form, which is given by \(y=a(x-h)^{2}+k\), where (h, k) are the coordinates of the vertex. Here, we can see that \(h = 2\) and \(k = 1\). Therefore, the vertex of the graph is at the point (2,1).
02

Identify the axis of symmetry

The axis of symmetry of a parabola in vertex form is given by the vertical line \(x=h\). In our case, the axis of symmetry is \(x=2\).
03

Determine the direction of the parabola

The coefficient "a" in the vertex form of the quadratic function determines the direction of the parabola. In our case, \(a = 1\) (since there is no coefficient explicitly given, it is understood to be 1). Since \(a > 0\), the parabola opens upwards.
04

Sketch the graph

Now that we have all the key features of the graph, we can sketch it. Begin by plotting the vertex at point (2,1). Draw the axis of symmetry at \(x=2\). Since the parabola opens upwards, sketch a curve centered at the vertex and opening in the upward direction. The graph of the function \(y=(x-2)^{2}+1\) is a parabola with vertex at (2,1), axis of symmetry at \(x=2\), and it opens upwards.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Parabolas
Graphing a parabola begins by understanding its equation, often given in vertex form: \[y = a(x - h)^2 + k.\]In this format,
  • \(h\) and \(k\) represent the coordinates of the vertex.
  • \(a\) is the coefficient that affects the direction of the opening and the "width" or "narrowness" of the parabola.
For the function \(y = (x - 2)^2 + 1\), the vertex \((h, k)\) is \( (2, 1) \). To graph this function, begin by plotting the vertex on a coordinate plane.
The vertex is the highest or lowest point on a parabola, depending on its direction. Add the axis of symmetry, a vertical line running through this point. Next, since the parabola is symmetrical over this line, plot additional points on each side of the vertex and mirror them. Finally, draw a smooth curve through these points. Remember, since we have a coefficient \(a = 1\), it maintains its standard width.
Axis of Symmetry
The axis of symmetry is a pivotal concept when working with parabolas, especially when graphing or analyzing their behavior.This "axis" denotes a line of symmetry that perfectly divides the parabola into two mirror-image halves.
For any parabola in vertex form, the axis of symmetry is directly tied to the vertex as:\[ x = h. \]For our function \(y = (x - 2)^2 + 1\), the axis of symmetry is \(x = 2\).
  • This means the parabola is evenly split across this imaginary line.
  • When graphing, this line aids in ensuring each side of the parabola mirrors the other.
By identifying this line, it becomes much easier to understand the parabola's structure and predict how it will appear on a graph.
Direction of Parabola
The direction in which a parabola opens is determined by the coefficient \(a\) in its vertex form \(y = a(x - h)^2 + k\).
  • When \(a > 0\), such as in our case where \(a = 1\), the parabola opens upwards, forming a "U" shape.
  • If \(a\) were less than zero \((a < 0)\), the parabola would open downwards, like an upside down "U".
The value of \(a\) also affects the "width" of the opening:
  • A larger absolute value of \(a\) makes the parabola narrower.
  • A smaller absolute value of \(a\), such as \(a = 1\), results in a standard, wider shape.
In essence, the sign and magnitude of \(a\) are crucial for predicting the general shape and direction of a parabola on a graph.

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