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Magazine Chapter 11: Problem 47
Sketch the graph of each quadratic function. See Section 8.5 $$ f(x)=(x-1)^{2}+3 $$
Short Answer
Expert verified
The graph is a parabola opening upwards with vertex at (1, 3) and symmetry along x=1.
Step by step solution
01
Identify the Vertex
The given quadratic function is in vertex form, which is \(f(x) = (x-h)^2 + k\). Here, \(h = 1\) and \(k = 3\). Therefore, the vertex of the parabola is \((1, 3)\).
02
Determine the Direction of Opening
Since the coefficient of \((x - 1)^2\) is positive (which is implicitly 1), the parabola opens upwards.
03
Find Additional Points
To sketch the graph accurately, find additional points. Choose \(x = 0\) and \(x = 2\) to find symmetrical points around the vertex. Calculate \(f(0) = (0-1)^2 + 3 = 4\) and \(f(2) = (2-1)^2 + 3 = 4\). Both points are \((0, 4)\) and \((2, 4)\).
04
Axis of Symmetry
The axis of symmetry for a parabola in vertex form \(f(x) = (x-h)^2 + k\) is \(x = h\). Thus the axis of symmetry for this function is \(x = 1\).
05
Sketch the Graph
Plot the vertex \((1, 3)\), and the points \((0, 4)\) and \((2, 4)\) on a coordinate plane. Draw a smooth curve that opens upwards and passes through these points, using \(x = 1\) as the line of symmetry.
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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 quadratic function is essential for easily identifying the key components of a parabola. The vertex form of a quadratic function is typically expressed as \(f(x) = (x-h)^2 + k\). Here, \(h\) and \(k\) represent the coordinates of the vertex of the parabola, which is a point on the graph where the curve changes direction.
- In our function \(f(x) = (x-1)^2 + 3\), we see that \(h = 1\) and \(k = 3\).
- This tells us the vertex is located at the point \((1, 3)\).
Parabola
A parabola is the U-shaped graph that represents a quadratic function. Its shape and orientation provide insights into the function's characteristics. Parabolas can open upwards or downwards depending on the sign of the quadratic term's coefficient.
- If the coefficient is positive, the parabola opens upwards.
- If the coefficient is negative, the parabola opens downwards.
Axis of Symmetry
The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves. For a quadratic function in vertex form, this line can be easily calculated. It runs through the vertex and is given by the formula \(x = h\).
- For the function \(f(x) = (x-1)^2 + 3\), the axis of symmetry is \(x = 1\).
- This line helps in plotting the graph accurately, as any point on the parabola on one side of this line has a corresponding point on the opposite side.
Graph Sketching
Graph sketching involves several steps that combine to provide a clear picture of the quadratic function's behavior. Starting with recognizing the vertex and the axis of symmetry, you can plot these points and use them as guides.
- Begin by plotting the vertex, in our case \((1, 3)\).
- Then, determine additional points by using symmetry. In this exercise, the points \((0, 4)\) and \((2, 4)\) are calculated and plotted.
