Textbook Question
The distance between the two points and is . Distance formula. Find the closest point on the line to the point . (Hint: Every point on has the form , and the closest point has the minimum distance.)
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4. Polynomial Functions
Quadratic Functions
Problem 23
Textbook Question
Graph each quadratic function. Give the (a) vertex, (b) axis, (c) domain, and (d) range. ƒ(x) = (x - 2)2
Verified step by step guidance1
Identify the given quadratic function: \(f(x) = (x - 2)^2\). This is in vertex form, which is \(f(x) = a(x - h)^2 + k\), where \((h, k)\) is the vertex.
Find the vertex by comparing \(f(x) = (x - 2)^2\) to the vertex form. Here, \(h = 2\) and \(k = 0\), so the vertex is at \((2, 0)\).
Determine the axis of symmetry, which is the vertical line that passes through the vertex. The axis is \(x = 2\).
State the domain of the function. Since this is a quadratic function, the domain is all real numbers, written as \((-\infty, \infty)\).
Find the range by considering the vertex and the direction the parabola opens. Since \(a = 1 > 0\), the parabola opens upward, so the range is \([0, \infty)\).
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vertex of a Quadratic Function
The vertex is the highest or lowest point on the graph of a quadratic function, representing its maximum or minimum value. For functions in the form f(x) = (x - h)^2 + k, the vertex is at (h, k). In this question, the vertex is at (2, 0), indicating the parabola opens upward from this point.
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Axis of Symmetry
The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves. It passes through the vertex and has the equation x = h for a quadratic in vertex form. Here, the axis of symmetry is x = 2, reflecting the parabola's symmetry about this line.
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Domain and Range of Quadratic Functions
The domain of any quadratic function is all real numbers since x can take any value. The range depends on the vertex and the parabola's direction; for f(x) = (x - 2)^2, the parabola opens upward with a minimum at y = 0, so the range is y ≥ 0.
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