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:

[table="width: 300, class: grid"]
[tr]
[td]\(x\)[/td]
[td]\(f(x)\)[/td]
[td]\(g(x)\)[/td]
[/tr]
[tr]
[td]-2[/td]
[td]24[/td]
[td]16[/td]
[/tr]
[tr]
[td]-1[/td]
[td]10[/td]
[td]10[/td]
[/tr]
[tr]
[td]0[/td]
[td]0[/td]
[td]6[/td]
[/tr]
[tr]
[td]1[/td]
[td]-6[/td]
[td]4[/td]
[/tr]
[tr]
[td]2[/td]
[td]-8[/td]
[td]4[/td]
[/tr]
[tr]
[td]3[/td]
[td]-6[/td]
[td]6[/td]
[/tr]
[tr]
[td]4[/td]
[td]0[/td]
[td]10[/td]
[/tr]
[tr]
[td]5[/td]
[td]10[/td]
[td]16[/td]
[/tr]
[tr]
[td]6[/td]
[td]24[/td]
[td]24[/td]
[/tr]
[tr]
[td]7[/td]
[td]42[/td]
[td]34[/td]
[/tr]
[/table]

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

$$f(x)=g(x)$$

$$2x^2-8x=x^2-3x+6$$

Collect everything on the LHS:

$$x^2-5x-6=0$$

Factor:

$$(x+1)(x-6)=0$$

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

$$x\in\{-1,6\}$$

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

$$f(-1)=2(-1)^2-8(-1)=2+8=10$$

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

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

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

And so we may conclude that the functions:

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

intersect at the points:

$$(-1,10)$$ and $$(6,24)$$

Does all that make sense?