1. Introduction
Recall that the graph of an equation, such as y = x + 1, is the complete set of solutions for that equation. The values for y are the result of a process being done to x. To make a general statement, we call the process by a letter, such as f, and we can call the results of that process “f of x.” We write “f of x” as f(x). The process f is a function; in a function, every element of the domain is paired with exactly one element of the range. That is, for every value of x, there is exactly one value of y. Both y and f(x) represent the outcome of the function f on x. So, for the equation y = x + 1, adding 1 to x is the function f. Therefore, f(x) is x + 1 and y = f(x).
3.1.2: Intersecting Graphs

2. Introduction, continued
Writing equations this way is called function notation. Function notation is a way to name a function using
f(x) instead of y. Using function notation, we can graph more than one function at a time. A set of more than one equation is called a system. If we call one function f and another g, then we can graph y = f(x) and y = g(x) on the same coordinate plane. By graphing the functions f and g on the same plane, we can more easily see the functions’ differences or similarities.
3.1.2: Intersecting Graphs

3. Introduction, continued
Functions can be named using any letter, though f and g are used often. In this lesson, we will work with pairs of equations, f(x) and g(x), attempting to find a solution set where f(x) = g(x). In other words, we want to find all values for x where f(x) and g(x) are the same.
3.1.2: Intersecting Graphs

4. Key Concepts Graphing Solutions of Functions
In the graph of the functions f(x) and g(x), the set of solutions for f(x) = g(x) will be where the two graphs intersect.
If f(x) = g(x) = b for a particular value of x, say a, then the point (a, b) will be on the curve defined by f and will also fall on the curve defined by g.
Since (a, b) falls on both curves, they must intersect at point (a, b).
3.1.2: Intersecting Graphs

5. Key Concepts, continued
In the graphs of f(x) and g(x), look for the point(s) where the curves intersect.
Substitute the x-value from the point into the functions to see if it is, or is close to, a solution.
Note that it is possible for a system of equations to intersect at more than one point, at only one point, or to not intersect at all.
3.1.2: Intersecting Graphs

6. Key Concepts, continued
Points of Intersection
More than one
One
None
Both points of intersection are solutions to the system.
The point of intersection is the system’s only solution.
There is no solution to the system.
3.1.2: Intersecting Graphs

7. Key Concepts, continued
Using a Table of Values to Find Solutions
Making a table of values for a system of two equations means listing the inputs for each equation and then comparing the outputs.
List the values of x to substitute into the first column.
List the first equation in the second column and the corresponding outputs, which are the resulting values after substituting in the x-values chosen in the first column.
3.1.2: Intersecting Graphs

8. Key Concepts, continued
List the second equation in the third column and the corresponding outputs, which are the resulting values after substituting in the x-values chosen in the first column.
In the fourth column, list the difference of the first functions’ outputs minus the second equations’ outputs.
Look for where the difference is the smallest in absolute value and for a sign change event.
3.1.2: Intersecting Graphs

9. Key Concepts, continued
A sign change event is where the values of f(x) – g(x) change from negative to positive (or positive to negative).
The solution(s) to the system are where the sign change occurs and the difference in the two outputs is smallest in absolute value.
If the difference between the outputs is 0, the value of x that was substituted is the x-coordinate of the solution, and the corresponding output is the
y-coordinate of the solution.
3.1.2: Intersecting Graphs

10. Common Errors/Misconceptions
believing that all graphs will cross
substituting values incorrectly
3.1.2: Intersecting Graphs

11. Guided Practice Example 2
Use a graph to approximate the solutions for the following system of equations. Find f(x) – g(x) for your estimates.
f(x) = 2x
g(x) = x + 2
3.1.2: Intersecting Graphs

12. Guided Practice: Example 2, continued Graph f(x) = 2x and g(x) = x + 2
on the same
coordinate plane.
3.1.2: Intersecting Graphs

13. Guided Practice: Example 2, continued
Approximate the values for x where
f(x) = g(x).
From the graph, –2 and 2 should be good estimates.
3.1.2: Intersecting Graphs

14. Guided Practice: Example 2, continued
Evaluate f(x) = 2x and g(x) = x + 2 for x = 2.
Change “f(x) =” and “g(x) =” to “y =” and substitute 2 for x.
y = 2x = 2(2) = 4
y = x + 2 = (2) + 2 = 4
3.1.2: Intersecting Graphs

15. Guided Practice: Example 2, continued
Find the difference in the y-values for the equations.
4 – 4 = 0; therefore, x = 2 satisfies f(x) = g(x) and is a solution to the system.
The point (2, 4) is a solution for both graphs.
3.1.2: Intersecting Graphs

16. Guided Practice: Example 2, continued
Evaluate f(x) = 2x and g(x) = x + 2 for x = –2.
Change “f(x) =” and “g(x) =” to “y =” and substitute –2 for x.
y = 2x = 2(–2) = 0.25
y = x + 2 = (–2) + 2 = 0
3.1.2: Intersecting Graphs

17. ✔ Guided Practice: Example 2, continued
Find the difference in the y-values for the equations.
0.25 – 0 = 0.25
0.25 is very close to 0; therefore, f(x) = g(x) when x is approximately equal to –2. ✔
3.1.2: Intersecting Graphs

18. Guided Practice: Example 2, continued 18
3.1.2: Intersecting Graphs

19. Guided Practice Example 3
Use a table of values to approximate the solutions for the following system of equations:
f(x) = 3x
g(x) = 2x + 1
3.1.2: Intersecting Graphs

20. Guided Practice: Example 3, continued Create a table of values.
f(x) = 3x
g(x) = 2x + 1
f(x) – g(x) –1
1.5
–1.17 1 2 3 9 5 4 27 18
3.1.2: Intersecting Graphs

21. ✔ Guided Practice: Example 3, continued
In column f(x) – g(x), look for sign changes.
There is a sign change from x = 0 to x = 2, and at
x = 1, f(x) – g(x) = 0. This tells us the curves f and g intersect at x = 1. ✔
3.1.2: Intersecting Graphs

22. Guided Practice: Example 3, continued 22
3.1.2: Intersecting Graphs