# intersection points of two quadratic functions

#### MattG03

##### New member
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

Staff member
Hello, and welcome to MHB!

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

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?