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Chapter 3.1. Quadratic Functions and Models. Example 1 Using Operations on Functions. Let f(x) = x 2 + 1 and g(x) = 3x + 5. The number a n is the leading coefficient of f(x). The function defined by f(x) – 0 is called the zero polynomial. The zero polynomial has no degree.
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Chapter 3.1 Quadratic Functions and Models
Example 1 Using Operations on Functions Let f(x) = x2 + 1 and g(x) = 3x + 5
The number an is the leading coefficient of f(x). The function defined by f(x) – 0 is called the zero polynomial. The zero polynomial has no degree. However, a polynomial function defined by f(x) = a0, for a nonzero number a0 has degree 0.
Quadratic Functions In sections 2.3 and 2.4, we discussed first-degree (linear) polynomial functions, in which the greatest power of the variable is 1. Now we look at polynomial functions of degree 2, called quadratic functions.
The simplest quadratic function id given by f(x) = x2 with a = 1, b = 0, and c = 0. To find some points on the graph of this function, choose some values for x and find the corresponding values for f(x), as in the table with Figure 1.
Then plot these points, and draw a smooth curve through them. This graph is called a parabola. Every quadratic function has a graph that is a parabola.
We can determine the domain and the range of a quadratic function whose graph is a parabola, such as the one in Figure 1, from its graph.
Since the graph extends indefinitely to the right and to the left, the domain is (-∞, +∞).
Since the lowest point is (0,0), the minimum range value (y-value) is 0.
The graph extends upward indefinitely; there is no maximum y-value, so the range is [0, ∞).
Parabolas are symmetric with respect to the line (the y-axis in Figure 1). The line of symmetry for a parabola is called the axis of the parabola.
The point where the axis intersects the parabola is called the vertex of the parabola.
As Figure 2 shows, the vertex of a parabola that opens down is the highest point of the graph and the vertex of a parabola that opens up is the lowest point of the graph.
Graphing Techniques The graphing techniques of Sections 2.6 applied to the graph of f(x) = x2 give the graph of any quadratic function.
The graph of g(x) = ax2 is a parabola with vertex at the origin that opens up if a is positive and downward if a is negative.
The width of the graph of g(x) is determined by the magnitude of a. That is, the graph of g(x) is narrower than that of f(x) = x2 if |a| > 1
The width of the graph of g(x) is determined by the magnitude of a. That is, the graph of g(x) is narrower than that of f(x) = x2 if |a| > 1 and is broader than that of f(x) = x2 if |a| < 1. By completing the square, any quadratic can be written in the form F(x) = a(x-h)2 + k.
The width of the graph of g(x) is determined by the magnitude of a. That is, the graph of g(x) is narrower than that of f(x) = x2 if |a| > 1 and is broader than that of f(x) = x2 if |a| < 1. By completing the square, any quadratic can be written in the form F(x) = a(x-h)2 + k.
The graph of F(x) is the same as the graph of g(x) = ax2 translated |h| units horizontally (to the right is h is positive and to the left is h is negative) and translated |k| units vertically (up if k is positive and down if is k is negative).
Example 1 Graph each function. Give the domain and the range y Graph each function x f(x) -1 x 0 1 2 3 4 5
Example 1 Graph each function. Give the domain and the range y Graph each function x f(x) -1 x 0 1 2 3 4 5
Think of g(x) = -½x2 as g(x) = -(½x2) . The graph of y = ½x2 is a broader version of y = x2,
Think of g(x) = -½x2 as g(x) = -(½x2) . The graph of y = ½x2 is a broader version of y = x2, and the graph of g(x) = -(½x2) is a reflection of y = ½x2 across the x-axis.
The vertex is (0,0), and the axis of the parabola is the line x = 0 (the y-axis). The domain is (- ∞, ∞), and the range is (- ∞, 0].
We can write F(x) = -½(x-4)2 +3 as F(x) = g(x-h)2 + k, where g(x) is the function of part (b), h is 4, and k is 3.
Therefore the graph of F(x) is the graph of g(x) granslated 4 usints to the right and 3 units up. See figure 7.
Therefore the graph of F(x) is the graph of g(x) granslated 4 usints to the right and 3 units up. See figure 7.
The vertex is (4,3), and the axis of the parabola is the line x = 4. The domain is (- ∞, ∞), and the range is (- ∞, 3].
Completing the Square In general, the graph of the quadratic function defined by f(x) = a (x – h)2 + k is a parabola with vertex (h, k) and axis x = h. The parabola opens up if a is positive and downward if a is negative.
With these facts in mind, we complete the square t graph a quadratic function defined by f(x) = ax2 + bx + c
The Vertex Formula We can generalize the earlier work to obtain a formula for the vertex of a parabola. Starting with the general quadratic form f(x) = ax2 + bx + c and completing the square will change the form to f(x) = a (x – h)2 + k
Comparing the last result with f(x) = ax2 + bx + c shows that
