How to Find Relative Minimum

Find the minimum point of a parabola by factoring and finding the roots.
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A relative minimum is the lowest point on a particular section of graph. This is in contrast to an absolute minimum, which is the lowest point on an entire graph. For example, you may be asked to find the relative minimum—the lowest point of the function—for cos(4x+1) between x=1 and x=2. This is asking you to consider only the area that is bounded by the values x=1 and x=2 in your calculations: the rest of the graph doesn't enter into the equation. Most graphing calculators give you the capability of graphing the function and finding relative minimums with the push of a button.

    Enter your function into the TI-89 graphing calculator. For example, if you were asked to find the minimum for cos(4x+1), press the green diamond button and the F1 key to reach the y= menu. Enter cos(4x+1) into y1 by pressing the following keys: 2nd Cos 4 x + 1 ).

    Graph the function. Press the green diamond and the F3 key.

    Press the F5 key to get the Math pull down menu.

    Use the down key to highlight 3:Minimum. Press enter.

    Select the lower bound by pressing the left arrow. When you have reached the far left of the area where you want to find the relative minimum (in our example, x=1), press enter.

    Select the upper bound by pressing the right arrow. When you have reached the far right of the area where you want to find the relative minimum (in our example, x=2), press enter.

    Press enter again. The calculator will display the Minimum x and y values. For the above example, the Minimum is x:.535, y:-1.


    • An alternative to using the arrow keys in Step 5 is to enter the x values before pressing the arrow key. For example, to enter the left value of x=1, press 1, then enter.


    • It's easy to get confused between absolute and relative minima. Make sure you read the question carefully and make sure you know which minimum you are calculating.

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