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May 19th

May 19th

# Finding a Vertex

 Consider the parabola to our left . It opens up. All parabolas with a positive leading coefficient open upwards. The leading coefficient is the number in front of x2. In the case to our left it is positive 1. Notice also that the vertex is the point with the smallest y value on the graph of the parabola . Mathematically we say it is the point with the Minimum y value. Before we start you should set your calculator to standard zoom by pressing: Now we will enter the equation. We will use the equation shown in the calculator. Press (arrow A). Use the clear button to remove any equations already present. Use the up and down arrow to place the curser to the right of the = sign. Enter the equation using the button (arrow B) for X. Use the button for the square exponent and use the subtraction symbol on the right of the calculator (arrow C) for the -6X since we are subtracting from X2. The key sequence should look like this : Now press the graph key (arrow D) to graph the parabola Your screen should look like the one at our right. Now press (arrow E) then (arrow F) Since our parabola is opening upward we want to find the x,y pair with the smallest (minimum ) y value. So chose 3 by pressing . First note the x=0 and y=13 at the bottom of the screen. This is the coordinate the curser is at. You can’t see it because at standard zoom the screen shows from -10 to +10 on both the x and y axis. Since the curser’s y coordinate is 13 the curser is above the top of the screen. Notice the Left Bound? Above the x=0. The calculator is asking for a left limit on the area it will search for a minimum. The left bound must be to the left of the vertex but not too far away. The left and right arrow keys move the curser along the graph of the equation shown at the top of the screen. The up and down arrow keys would jump the curser to another curve if a second or more equation had been plotted. Use the right arrow (arrow g) to move the curser to about the same place shown in the picture at left. Notice the curser is to the left of the vertex marked with the red circle. (I added this red circle, it will not appear on your screen) This is a good left bound. Press to accept this left bound. Notice the question at the bottom of the screen has changed to asking for a right bound. Use the right arrow to move the curser to the position shown at left. Notice that this curser is located to the right of the vertex marked with the red circle. This is a good right bound. Press to accept this right bound. Notice the two opposing arrows on the screen at arrow h. The calculator will search x values between the two arrows for the minimum y value. If the vertex is not between these two arrow is will not find it. Notice the question at the bottom of the screen is asking for a guess. The guess is the point the calculator will start at searching for the vertex. Use the left arrow (arrow I) to move the curser back to the vertex as shown at left. The curser at left is a good guess. Press to accept this guess. Notice there is Minimum displayed in the bottom left of the screen and x=2.99999999 y=4. I would interpret the x=2.9999999 to be x=3 We need to look at a parabola opening downward. Press and to remove the first equation. Enter the equation at left. Notice the negative in front of X2 is the key at arrow J. Use this key when there is nothing to subtract from. Use the minus at arrow K in front of the 5 since it is being subtracted from 6X. Once the equation is in press . Your screen should look like the one be low :Notice the parabola is opening downward. The vertex is not the smallest y value but the largest. We need to find the largest y value on the graph of the equation. In mathematics this is called the maximum y value. Press . You should see the screen at the left. Press to chose the maximum function. Notice the question is what is the left bound and the curser is at x=0, y=-5. Use the right arrow (arrow K) to move the curser to the point shown at left. This position is relatively close but definitely to the left of the vertex marked with the red circle. This makes it a good left bound so press to accept this left bound. Notice the question has changed to Right Bound? Asking for a right bound of the area to be searched for the maximum. Use the right arrow (arrow L) to move the curser to the point shown at left. Notice this point is relatively close to the vertex marked with the red circle but definitely to the right of the vertex. This makes it an appropriate right bound. Press to accept the right bound. Notice the two arrows at arrow M. The calculator will search for the maximum between these two arrows. If the vertex is not between these two arrows it will not find it. Notice the question in the bottom left of the screen has changed to Guess? Move the curser left to the point shown using the left arrow (arrow N). Notice the guess is as close to the vertex marked with the red circle as possible. Press to accept this guess. The vertex’ x coordinate is located at x=2.9999978 (this is an approximation and you must use your judgment to decide whether or not to round this to 3) The vertex’ y coordinate is at y=4. Remember if the parabola opens up use the minimum function and if it opens down use the maximum function. To graph parabola’s it is useful to find the x intercepts .
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