Intersection points of two quadratic functions

In summary, we used a graph, table, and algebraic equations to justify the given points (-1,10) and (6,24) for the functions f(x) = 2x^2 - 8x and g(x) = x^2 - 3x + 6. The graph showed that the points were the intersections of the two functions, the table showed that the values of x for which the functions are equal are -1 and 6, and the algebraic equations verified that the points indeed satisfy the equations.
  • #1
MattG03
1
0
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)
 
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  • #2
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:

\(x\)\(f(x)\)\(g(x)\)
-22416
-11010
006
1-64
2-84
3-66
4010
51016
62424
74234

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?
 

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Related to Intersection points of two quadratic functions

1. What are the intersection points of two quadratic functions?

The intersection points of two quadratic functions are the points where the two functions intersect or cross each other on a graph. These points represent the values of x and y where both functions have the same output.

2. How do I find the intersection points of two quadratic functions?

To find the intersection points of two quadratic functions, set the two functions equal to each other and solve for x. The resulting values of x will be the x-coordinates of the intersection points. To find the y-coordinates, substitute the x-values into either of the original functions.

3. Can two quadratic functions have more than two intersection points?

Yes, two quadratic functions can have more than two intersection points. In fact, if the two functions have different leading coefficients and/or different constants, they can intersect at up to four different points.

4. What does it mean if two quadratic functions have no intersection points?

If two quadratic functions have no intersection points, it means that the two functions do not intersect or cross each other on a graph. This could mean that the functions are parallel or that they have no real solutions.

5. How can I use the intersection points of two quadratic functions in real-life applications?

The intersection points of two quadratic functions can be used in many real-life applications, such as finding the break-even point for a business, determining the optimal location for a bridge, or calculating the maximum or minimum values of a problem. These points represent important values that can help make informed decisions in various situations.

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