Taylor graphs the system below on her graphing calculator and decides that f(x)=g(x) at x=0, x=1, and x=3.

Question

1.Your friend, Taylor, missed class today and needs some help identifying solutions to systems. Explain to Taylor how to find the solution(s) of any system using its graph.

2.Taylor graphs the system below on her graphing calculator and decides that f(x)=g(x) at x=0, x=1, and x=3. Provide Taylor some feedback that explains which part of her answer is incorrect and why it is incorrect.

f(x)=2x+1

g(x)=2x^2+1

Answer 1

f(x) = 2x + 1

g(x) = 2x² + 1

Set f(x) = g(x) ⇒ 2x + 1 = 2x² + 1

We want to rewrite the equation in quadratic form. Then factor it using GCF (greatest common factor). Apply Zero Product Property to set both factors equal to 0 to find x.

2x² - 2x = 0 ⇒ 2x(x - 1) = 0 ⇒ 2x = 0 or x -1 = 0 ⇒ x = 0 or x = 1

Plug in the x values to find y in order to create the solution points for the system of equations.

f(0) = 2(0) + 1 = 0 + 1 = 1

f(1) = 2(1) + 1 = 2 + 1 = 3

g(0) = 2(0)² + 1 = 2(0) + 1 = 0 + 1 = 1

g(1) = 2(1)² + 1 = 2(1) + 1 = 2 + 1 = 3

The solution points are (0, 1) and (1, 3). These are the intersection points of the two equations.

Lets see if x =3, then we have:

f(3) = 2(3) + 1 = 6 + 1 = 7

g(3) = 2(3)² + 1 = 2(9) + 1 = 18 + 1 = 19

f(3) ≠ g(3) ⇒ 7 ≠ 19

3 is not part of the solution to the system of equations.