Solve the equation resulting from equating f(x) and g(x). The solutions are the x-coordinates of the points of intersection.
(3, 10) and (1/2, 0 )
Practice makes perfect
We want to find the points at which the graphs of functions f(x) = 2x^2-3x+1 and g(x) = 4x-2 intersect. If x is an x-coordinate of the point of intersection, then f(x) = g(x). Therefore, we can set these functions equal to each other to write an equation for such values of x.
f(x)&=g(x)
2x^2-3x+1&=4x-2
By doing so, we have written a quadratic equation for x. Now, let's transform it so that the right-hand side equals 0.
Now, by the Zero Product Property the resulting equation is satisfied if either of the two factor equals 0.
(x-3)(2x-1) = 0 ⇕ x-3 = 0 or 2x-1 = 0
Let's solve the first of the resulting equations.
x- 3 = 0 ⇔ x = 3
One of the solutions to our equation is x=3. Now, let's find the other solution by solving the second equation.
The second solution to our equation is x = 12. The two solutions to our equation are also the x-coordinates of the points of intersection of our functions. To find the y-coordinates by substituting each solution for x in the equation for g(x), one solution at a time. We will start with x=3.
