Question

Consider the graph of the functions (x-5)^2and = -3x+13. What are the points at which f(x)=g(x)?

Answer 1

$\begin{gathered} f(x)=(x-5)^2 \\ g(x)=-3x+13 \end{gathered}$

To find the points at which the functions are equal:

1. Equal f to g:

$\begin{gathered} f(x)=g(x) \\ \\ (x-5)^2=-3x+13 \end{gathered}$

2. Solve x:

$\begin{gathered} (a-b)^2=a^2-2ab+b^2 \\ \\ x^2-10x+25=-3x+13 \\ x^2-10x+25+3x-13=0 \\ x^2-7x+12=0 \\ \\ ax^2+bx+c=0 \\ x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \\ \\ x=\frac{-(-7)\pm\sqrt[]{(-7)^2-4(1)(12)}}{2(1)} \\ x=\frac{7\pm\sqrt[]{49-48}}{2} \\ x=\frac{7\pm\sqrt[]{1}}{2} \\ x=\frac{7\pm1}{2} \\ \\ x_1=\frac{7+1}{2}=\frac{8}{2}=4 \\ \\ x_2=\frac{7-1}{2}=\frac{6}{2}=3^{} \end{gathered}$

Then, f and g are equal when x=3 and x=4. Find the corresponding values of y:

$\begin{gathered} f(3)=(3-5)^2=(-2)^2=4 \\ \\ f(4)=(4-5)^2=(-1)^2=1 \end{gathered}$

Then, the points at which the functions are equal are: (3,4) and (4,1)