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

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).

SIGH

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)

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.

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

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 -15 + 2 = -6 – 7 -13 = -13

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)?

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

What if the two graphs do not intersect???

Now let’s look at some you did last night… http://www.uncwil.edu/courses/mat111hb/calculators/graphcalc.html