Intersection points of two quadratic functions

[MattG03]

Justify the following by using table, graph and equation. use words to explain each representation
f(X) = 2 x2 - 8x and g(x) = x2-3x+ 6 the points (-1,10) and (6,24)

 

[MarkFL]

Hello, and welcome to MHB! (Wave)

Let's first look at a graph of the two given functions and the given points:

View attachment 8479

From the graph, it appears the given points are where the two functions \(f\) and \(g\) intersect. Let's look at a table:

\(x\)	\(f(x)\)	\(g(x)\)
-2	24	16
-1	10	10
0	0	6
1	-6	4
2	-8	4
3	-6	6
4	0	10
5	10	16
6	24	24
7	42	34

The table would indicate that the two given points are where the two functions intersect. Now, we can verify this algebraically as follows:

\(\displaystyle f(x)=g(x)\)

\(\displaystyle 2x^2-8x=x^2-3x+6\)

Collect everything on the LHS:

\(\displaystyle x^2-5x-6=0\)

Factor:

\(\displaystyle (x+1)(x-6)=0\)

And so the values of \(x\) for which the two functions are equal are:

\(\displaystyle x\in\{-1,6\}\)

Let's verify by finding the values of the functions for those two values of \(x\):

\(\displaystyle f(-1)=2(-1)^2-8(-1)=2+8=10\)

\(\displaystyle g(-1)=(-1)^2-3(-1)+6=1+3+6=10\)

\(\displaystyle f(6)=2(6)^2-8(6)=72-48=24\)

\(\displaystyle g(6)=(6)^2-3(6)+6=36-18+6=24\)

And so we may conclude that the functions:

\(\displaystyle f(x)=2x^2-8x\) and \(\displaystyle g(x)=x^2-3x+6\)

intersect at the points:

\(\displaystyle (-1,10)\) and \(\displaystyle (6,24)\)

Does all that make sense?