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Use graphing technology to graph a function that matches each given graph. Make sure that your graph goes through all 3 points shown!
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Equation:
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In an earlier lesson, we saw that a quadratic function written in standard form, , can tell us some things about the graph that represents it. The coefficient can tell us whether the graph of the function opens upward or downward, and also gives us information about whether it is narrow or wide. The constant term can tell us about its vertical position.
Recall that the graph representing is an upward-opening parabola with the vertex at . The vertex is also the -intercept and the -intercept.
Suppose we add 6 to the squared term: . Adding a 6 shifts the graph upward, so the vertex is at . The vertex is the -intercept, and the graph is centered on the -axis.
What can the linear term tell us about the graph representing a quadratic function?
The linear term has a somewhat mysterious effect on the graph of a quadratic function. The graph seems to shift both horizontally and vertically. When we add (where is not 0) to , the graph of is no longer centered on the -axis.
Suppose we add to the squared term: . Writing the in factored form as gives us the zeros of the function, 0 and -6. Adding the term seems to shift the graph to the left and down and the -intercepts are now and . The vertex is no longer the -intercept, and the graph is no longer centered on the -axis.
What if we add to ? We know that can be rewritten as , which tells us the zeros: 0 and 6. Adding a negative linear term to a squared term seems to shift the graph to the right and down. The -intercepts are now and . The vertex is no longer the -intercept, and the graph is not centered on the -axis.