TI-NSpire Manual

Key Features of Graphs

Find the maximum or minimum of a function

Suppose you want to know the local maximums and minimums of the following function:

\[ f(x) = 2x^3 - 9x^2 + 12x - 3 \]

Enter the function and graph it properly. Here we choose Xmin= 0, Xmax= 3, Ymin= 0, Ymax= 4:
Press and select Analyze Graph > Maximum.
Choose a left bound (as asked at the bottom of the screen) until you reach the left of the local maximum:
Press and select Analyze Graph > Maximum.
Press
Choose a right bound (as asked at the bottom of the screen):
Press
Read the maximum:
Press .
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.

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 above.

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

\[ f(x) = x^3 - x + 1, \quad -1 \leq x \leq 1. \]

Input the function
¨ Press and select Window / Zoom > Window Settings and fill Xmin and Xmax accordingly to the domain of the function:
¨ Press and select Window / Zoom > Zoom - Fit, and wait. The following should display:
¨ Press and select Window / Zoom > Window Settings again, and read the values of Ymin and Ymax:

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

\[ 0.538 \leq y \leq 1.462. \]

Suppose you want to know the local maximums and minimums of the following function:

\[ f(x) = 2x^3 - 9x^2 + 12x - 3 \]

Recall that we want to compute f (0).

Enter the function, and display its graph.
Press and select Trace > Graph Trace. Press and it automatically shows the y−intercept.

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

Suppose you want to know the local maximums and minimums of the following function:

\[ f(x) = 2x^3 - 9x^2 + 12x - 3 \]

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

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

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

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

\[ \frac{2x^3 - 4x + 2}{5x^3 - x^2 + 2} \]

Enter the function and display huge values of x, like Xmax=10000:
Press , select Analyze Graph > Maximum and take the y value of the maximum as the value of the horitzontal asymptote:

The result should be Y=0.4 (rounded).

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

\[ \frac{2x^3 - 4x + 2}{5x^3 - x^2 + 2} \]

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:
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).
Take the average of the two values.
The result should be x = 0.676 (rounded).

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).

Suppose you want to know one of the intersections of the graphs of the following functions:

\[ f(x) = x^2 - 2x + 2, \quad g(x) = \frac{x + 10}{4}. \]

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

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