# Finding Intersections

An intersection point is where two or more graphs coincide. It is a point that is the solution to a system of equations. The TI series is limited to the intersection of two curves.

## All calculators

1. Solve the equation for Y if it isn't already that way.
2. Go to Y=
3. Enter the first function into y1
4. Enter the second function into y2
5. Hit Graph
6. Change the viewing window if necessary so that you can see where the graph has an intersection. You need to make sure that there is enough to the left and right of the intersection to select a point.
7. Continue on to the specific steps for your calculator. If there are additional intersections to find, you may need to change your viewing window and repeat the steps for each intersection.

## TI-82

When selecting the two curves to use, the TI-82 defaults to y1 and y2, so if those are the curves you're using, you just hit enter when it asks for the first and second curves. If your curves are not in y1 or y2, then you can use the up or down arrow keys to select the different curves. The TI-82 will show you a 1 or 2 for the function number in the upper left hand corner of the screen.

1. Press Calc (2nd Trace)
2. Choose Intersect (#5)
3. For the First Curve, hit enter.
4. For the Second Curve, hit enter
5. For the Guess, arrow to the intersection and press enter.
6. The TI-82 will return a value for x and y. Both of these coordinates are part of the solution.

## TI-83

When selecting the two curves to use, the TI-83 defaults to y1 and y2, so if those are the curves you're using, you just hit enter when it asks for the first and second curves. If your curves are not in y1 or y2, then you can use the up or down arrow keys to select the different curves. The TI-83 will show you the equation being graphed in the upper left hand corner of the screen.

1. Press Calc (2nd Trace)
2. Choose Intersect (#5)
3. For the First Curve, hit enter.
4. For the Second Curve, hit enter
5. For the Guess, arrow to the intersection and press enter.
6. The TI-83 will return a value for x and y. Both of these coordinates are part of the solution.

## TI-85 / TI-86

When selecting the two curves to use, the TI-85 defaults to y1 and y2, so if those are the curves you're using, you just hit enter when it asks for the first and second curves. If your curves are not in y1 or y2, then you can use the up or down arrow keys to select the different curves. The TI-85 doesn't let you know that it's looking for the curves, it just shows a 1 or 2 for the curve number in the upper right hand corner of the screen.

1. Press Graph
2. Press More and then Math (F1)
3. Press Lower (F1). Arrow to the left of the intersection and press enter.
4. Press Upper (F2). Arrow to the right of the intersection and press enter.
5. Press More and then ISect (F5)
6. Hit enter to select the first curve.
7. Arrow to a point close to the intersection and hit enter to select the second curve.
8. The TI-85 will return a value for x and y. Both of these coordinates are part of the solution.

You do not need to supply a guess for the TI-85. You should make sure there is only one intersection in the interval specified or you won't know which one it is going to give you. You can provide a hint for the TI-85 by arrowing towards the intersection point you want.

## TI-89 / TI-92

When selecting the two curves to use, the TI-89 defaults to y1 and y2, so if those are the curves you're using, you just hit enter when it asks for the first and second curves. If your curves are not in y1 or y2, then you can use the up or down arrow keys to select the different curves. The TI-89 will show you the equation being graphed in the upper left hand corner of the screen

1. Press Math (F5)
2. Choose Intersection (#5)
3. For the 1st Curve, press enter.
4. For the 2nd Curve, press enter.
5. For the Lower Bound, arrow to the left of the intersection and hit enter.
6. For the Upper Bound arrow to the right of the intersection and hit enter.
7. The TI-89 will return a value for x and y. The x is where the intersection occurs and the y is the minimum or maximum value.

You do not need to supply a guess for the TI-89. You should make sure there is only one intersection in the interval specified or you won't know which one it is going to give you.

## Example (TI-83+)

Find the intersection of the two curves y = ex and y=4-x2

Enter the y1 = ex and y2 = 4 - x2 and then graph. You may need to change your viewing window so that the intersection is visible. My window settings for the graphs below are XMin = -3, XMax = 3, YMin = -1, and YMax = 5.

We'll find the intersection on the left, first.

Graph Zero Curve 1 Curve 2 Guess Intersection

The calculator says the solution is x = -1.964636 and y = 0.14020697

Repeat the steps to find the other intersection point.

Guess Intersection

The second intersection point is at x = 1.0580064 and y = 2.8806225.

Note that you do not have to have all of the intersections showing on the screen at one time. You can change the viewing window to find other intersection points. This is true of all features of the calc menu. Only the part of the graph that you're interested in needs to be visible. Other x-intercepts, maximums, minimums, or intercepts don't have to be showing.