The simplest form of a quadratic function is f (x) x2. Let’s graph this function as well as a similar function, g(x) x2 2. EXAMPLE 1 Graph f (x) x2 and g(x) x2 2 on the same axes. Solution We will make a table of values for each function, then plot the points. Axis of Symmetry y 7 up 2 up 2 Vertex g(x) x2 2 f(x) x2 5 5 3 Vertex f(x) x2 x f(x) 0 0 1 1 2 4 1 1 2 4 g(x) x2 2 x g(x) 0 2 1 3 2 6 1 3 2 6 The domain of f(x) is (q, q), and the range is 0, q). The domain of g(x) is (q, q), and the range is 2, q). Definition The graph of a quadratic function is called a parabola. The lowest point on a parabola that opens upward or the highest point on a parabola that opens downward is called the vertex. The vertex of the graph of f (x) in Example 1 is (0, 0), and the vertex of the graph of g(x) is (0, 2). Every parabola has symmetry. Let’s look at the graph of f (x) x2. If we were to fold the paper along the y-axis, one half of the graph of f (x) x2 would fall exactly on the other half. The y-axis, or the line x 0, is the axis of symmetry of f (x) x2. (It is also true that the line x 0 is the axis of symmetry of g(x) x2 2.) We can see from the tables of values in Example 1 that although the x-values are the same in each table, the corresponding y-values in the table for g(x) are 2 more than the y-values in the fi rst table. In other words, if f (x) x2, then g(x) x2 2, so g(x) f (x) 2. The y-coordinates of the ordered pairs of g(x) are 2 more than the y-coordinates of the ordered pairs of f (x) when the ordered pairs of f and g have the same x-coordinates. This means that the graph of g is the same shape as the graph of f, but g is shifted up 2 units. We can make the following general statement about shifting the graph of a function vertically. Property Vertical Shifts Given the graph of f (x), if g(x) f (x) k, where k is a constant, then the graph of g(x) is the same shape as the graph of f (x) but g is shifted vertically k units. x In your own words, define the vertex of a parabola, the axis of symmetry, and vertical shift. 654 CHAPTER 10 Quadratic Equations and Functions www.mhhe.com/messersmith
messersmith_power_intermediate_algebra_1e_ch4_7_10
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