1 Find the maximum or minimum of a function, and its range

Find a local maximum or minimum

Suppose you want to know the local maximum(s) and minimum(s) of the following function:

f (x ) = 2x3 - 9x2 + 12x - 3

①
Enter the function and graph it properly. Here we choose Xmin= 0, Xmax= 3, Ymin= 0, Ymax= 4:

PIC

②
Press PIC and select Analyze Graph > Maximum.
③
Choose a left bound (as asked at the bottom of the screen) until you reach the left of the local maximum:

PIC
Press PIC

④
Choose a right bound (as asked at the bottom of the screen):

PIC
Press PIC

⑤
Read the maximum:

PIC
Press PIC

The result should be x = 1 and y = 2 (it is displayed next to the point).

You can apply the same procedure to find a local minimum by pressing select Analyze Graph > Minimum instead of select Analyze Graph > Maximum.

Find together the range, the global maximum and the global minimum of a function within a domain

PIC

This strategy is useful when one one only wants the y values of the global maximum or minimum. If you have to know the x values too, see 1

Suppose you want to find the maximum and minimum of the following function:

3 f (x ) = x - x + 1, - 1 ≤ x ≤ 1.

①
Input the function
②
Press PIC and select Window / Zoom > Window Settings and fill Xmin and Xmax accordingly to the domain of the function:

PIC

③
Press PIC and select Window / Zoom > Zoom - Fit, and wait. The following should display:

PIC

④
Press PIC and select Window / Zoom > Window Settings again, and read the values of Ymin and Ymax:

PIC

Thus, the minimum of f is 0.538, and the maximum is 1.462. Thus, the range of f is

0.538 ≤ y ≤ 1.462.

2 Find the y-intercept of a function

Suppose you want to know the y-intercept of the following function:

3 2 f (x ) = 2x - 9x + 12x - 3

Recall that we want to compute f(0).

①
Enter the function, and display its graph.
②
Press PIC and select Trace > Graph Trace. Press PIC and it automatically shows the y-intercept.

PIC

The result Y=-3 should be displayed within the coordinate of the point (at the bottom right of the screen).

3 Find one of the x-intercepts of a function

Suppose you want to know the x-intercept of the following function:

f (x ) = 2x3 - 9x2 + 12x - 3

Recall that we want to compute the value of x when f(x) = 0.

①
Enter the function, and display its graph.
②
Press PIC and select Analyze Graph > Zero. Select lower and upper bounds the widest possible. Press PIC .

PIC

The result X=0.322 should be displayed within the coordinate of the point.

4 Find a horizontal asymptote of a function

Suppose you want to see the horizontal asymptote of the following function when x goes to ∞ :

2x3----4x +-2 5x3 - x2 + 2

①
Enter the function and display huge values of x, like Xmax=10000:

PIC

②
Press PIC , select Analyze Graph > Maximum and take the y value of the maximum as the value of the horitzontal asymptote:

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The result should be Y=0.4 (rounded).

5 Find a vertical asymptote of a function

Suppose you want to see a vertical asymptote of the following function:

2x3 - 4x + 2 ---3----2---- 5x - x + 2

①
Enter the function and display x-values near the vertical asymptote and big y-values, here Xmin=-1, Xmax=-0.5, Ymin=-10000 and Ymax=10000:

PIC

②
Use the maximum and minimum to find the x-value of each vertical asymptote. The x-values should be X=-0.678 at the left and X=-0.674 at the right (rounded).
③
Take the average of the two values.

The result should be x = 0.676 (rounded).

PIC

Depending on how precise you want the result to be, you will have to zoom in more, and move again the cursor

Here, by choosing Xmin=-0.7 and Xmax=-0.6 we get the result x = 0.67613 (rounded).

6 Find the intersection of two curves

Suppose you want to know one of the intersections of the graphs of the following functions:
f(x) = x2 - 2x + 2 g(x) = x-+-10- 4.

①
Enter the two functions and graph them in order to see the intersection in question.
②
Press PIC and select Analyze Graph > Intersection.
③
Select right and left bound according to the intersection you want to know between the two intersections (right or left).
④
Press PIC . The intersection point coordinates are displayed:

PIC
Press PIC

The intersection points should be (-0.204,2.45) (left intersection point) and (2.45,3.11) (right intersection point) (rounded).

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