E1.06: Graphs Part 2
Example 9. Just by sketching some more of the graph, estimate the yvalue of [latex]y=4+2{{(x3)}^{2}}[/latex] when [latex]x=14[/latex]. Solution: Extend the graph a bit and find that it appears to give [latex]y=250[/latex]. Check: Check this by plugging [latex]x=14[/latex] into the formula, [latex]y=4+2{{(x3)}^{2}}=4+2{{(143)}^{2}}=4+2{{(11)}^{2}}=246[/latex] Example 10. Use the graph to estimate which x gives the lowest value for y when [latex]y=4+2{{(x3)}^{2}}[/latex] on the values [latex]0\le{x}\le12[/latex]. Solution: That xvalue is clearly between 0 and 5. It appears to be a bit above halfway. So we estimate that it is about [latex]x=3[/latex]. Example 11. Let’s “magnify” the portion of the graph near [latex]x=3[/latex] in order to see very precisely where the minimum value is. Actually, we leave the old dataset and graph alone and produce a new one. This time, we’ll just use x values near 3. So we’ll graph [latex]y=4+2{{(x3)}^{2}}[/latex] on the values [latex]2\le{x}\le4[/latex] where we increase the xvalues in increments of 0.1. Use the same technique as before. Here is the middle part of the data table and the graph. This makes clear that [latex]x=3[/latex] gives the minimum value for y.

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 Mathematics for Modeling. Authored by: Mary Parker and Hunter Ellinger. License: CC BY: Attribution.