1. Solving Equations f(x) = g(x)
Graphically

2. Yesterday we looked at solving equations where f(x) = g(x) by putting f(x) equal to g(x) and solving for x.
Today we will solve the equations where f(x) = g(x) by looking at their graphs.
This process is very similar
to solving a system of equations
(like you did in 8th grade).

3. SIGH

4. When two equations are graphed, there is a possibility that they may intersect.
If they do intersect, the point of intersection will give the x-value that makes BOTH equations equal to the same value or equal to each other...
f(x) = g(x)

5. When we have an equation to solve graphically…
Take each side of the equal sign and set equal to y (one will be f(x) the other g(x)) Graph both new y = equations on the same graph. The x-value(s) of the point(s) of intersection is the solution to the equation.

6. Example 1: Solve 5x + 2 = 2x - 7 f(x): y = 5x + 2 g(x): y = 2x – 7
Let’s look at the
graphs of the two
functions:
The x-value of the
point of intersection is…
Thinking back from our function families, what kind of functions are f(x) and g(x)?
x = -3

7. Let’s check our answer.
5x + 2 = 2x - 7
We should be able to substitute x = -3 and it will make both sides of the equal sign EQUAL to each other…
5(-3) + 2 = 2(-3) – 7
= -6 – 7
-13 = -13

8. What kind of functions are f(x) and g(x)?
Example 2: Solve: x2 + 1 = x + 3
f(x): y = x2 + 1
g(x): y = x + 3.
2. Graph:
3. The graphs intersect
in two points, and
the x-coordinates of
the intersection points are -1 and 2.
What kind of functions are f(x) and g(x)?

9. Two solutions???? Let’s check: x2 + 1 = x + 3
1st solution: x = -1 (-1)2 + 1 = = 2 2 = 2
2nd solution: x = 2
(2)2 + 1 = 2 + 3
4 + 1 = 5
5 = 5
Both work!

10. What if the two graphs do not intersect???

11. Now let’s look at some you did last night…