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Magazine Chapter 9: Problem 27
Graph each parabola. Plot at least two points as well as the vertex. Give the vertex, axis, domain, and range . $$ f(x)=(x-4)^{2} $$
Short Answer
Expert verified
Vertex: (4, 0), Axis of Symmetry: x = 4, Domain: (-鈭�, 鈭�), Range: [0, 鈭瀅
Step by step solution
01
Identify the Vertex
The given function is in vertex form, \( f(x) = (x-4)^2 \). We can see that the vertex form is \( f(x) = (x-h)^2 + k \). Here, \( h = 4 \) and \( k = 0 \), so the vertex is at (4, 0).
02
Axis of Symmetry
The axis of symmetry is a vertical line that goes through the vertex. For the vertex (4, 0), the axis of symmetry is the vertical line \( x = 4 \).
03
Plotting Points
Choose two points to plot on either side of the vertex. For example, \( x = 3 \) and \( x = 5 \) are one unit away from the vertex. We calculate the corresponding function values.For \( x = 3 \): \[ f(3) = (3 - 4)^2 = (-1)^2 = 1 \]So, the point (3, 1) lies on the parabola.For \( x = 5 \): \[ f(5) = (5 - 4)^2 = 1^2 = 1 \]So, the point (5, 1) lies on the parabola.
04
Determine the Domain
The domain of the function \( f(x) = (x-4)^2 \) is the set of all real numbers because there are no restrictions on \( x \). In set notation, the domain is \( (-fty, fty) \).
05
Determine the Range
The range of the function \( f(x) = (x-4)^2 \) is the set of all non-negative real numbers because the square of a real number is always non-negative. So the range is \( [0, fty) \).
06
Graph the Function
Plot the vertex (4, 0) and the points (3, 1) and (5, 1). Draw a smooth curve through these points to form the parabola. The graph will open upwards.
07
Summarize Key Features
Vertex: (4, 0)Axis of Symmetry: \( x = 4 \)Domain: \( (-fty, fty) \)Range: \( [0, fty) \)
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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 because it provides key information about the graph's highest or lowest point. In the equation given, we can see it follows the vertex form of a quadratic function: \( f(x) = (x - h)^2 + k \). This form makes it easy to identify the vertex. Here, the values of \( h \) and \( k \) are 4 and 0 respectively. Therefore, the vertex of the parabola described by the function \( f(x) = (x - 4)^2 \) is at the point (4, 0). The vertex tells us where the graph changes direction and is the minimum point because the parabola opens upwards.
axis of symmetry
The axis of symmetry is a vertical line that divides the parabola into two mirror images. This line always passes through the vertex. For our parabola, since the vertex is at (4, 0), the axis of symmetry is the line \( x = 4 \). Drawing this line helps us in graphing the parabola accurately because each point on one side of the axis has a corresponding point on the other side. To visualize it, imagine a standing mirror placed along the line \( x = 4 \). The left side of the parabola would reflect perfectly onto the right side and vice versa.
domain and range of functions
Determining the domain and range of a function helps us understand the set of possible inputs (domain) and outputs (range). For the function \( f(x) = (x-4)^2 \), the domain is all real numbers. There are no restrictions on the values that \( x \) can take, which in set notation is written as \( (-鈭�, 鈭�) \).
When we look at the range, we consider the possible values of \( f(x) \). Since the function is a square of \( (x-4) \) and squaring any real number always produces a non-negative result, the function can only produce values from 0 upwards. Thus, the range is \( [0, 鈭�) \).
When we look at the range, we consider the possible values of \( f(x) \). Since the function is a square of \( (x-4) \) and squaring any real number always produces a non-negative result, the function can only produce values from 0 upwards. Thus, the range is \( [0, 鈭�) \).
plotting points on a parabola
Plotting points is an essential step in graphing a parabola. We already have the vertex and axis of symmetry. Now we plot additional points to define the shape. Choosing values of \( x \) on either side of the vertex allows us to draw a more accurate graph. For example, choosing \( x = 3 \) and \( x = 5 \), we calculate:
Plot these points, and connect them smoothly with the vertex to form the parabolic curve. These steps ensure we can see the symmetric and curved nature of the graph accurately. For better precision, you can plot more points if needed.
- \( f(3) = (3 - 4)^2 = 1 \)
- \( f(5) = (5 - 4)^2 = 1 \)
Plot these points, and connect them smoothly with the vertex to form the parabolic curve. These steps ensure we can see the symmetric and curved nature of the graph accurately. For better precision, you can plot more points if needed.
