Algebra Graph problem, please help!

Question

1. Use what you have learned about quadratic functions to graph the following quadratic function on a coordinate grid.

f(x)=‒x²+6x‒9

Then, use what you have learned about the inverse functions to graph the inverse of the function.

Write three ordered pairs that lie on the inverse of the function.

   

2. Kayla and Jody are working at a station. They have been given four cards with four different functions on them. 

Function A	Function B	Function C	Function D
f(x)=2x+1	f(x)=2(x‒1)2	f(x)=2x+2	f(x)=2x2‒1

 

Kayla states that Function D always exceeds the other functions. Jody doesn't think Kayla's statement is true. Which values of x disproves her claim? Explain your reasoning.

Answer 1

f(x)= - x^2 +6x -9

f(x) = -(x^2 -6x +9) = -(x-3)^2 is a downward opening parabola with vertex (3,0) = maximum point

y intercept = (0,-9)

axis of symmetry is x=3

another point on the parabola is (6,-9)

connect the three points with a smooth parabolic curve

3 ordered pairs on the graph are (3,0), (0,-9) and (6,-9)

the inverse function is found by switching x and y and solving for the new y

x =-y^2 +6y -9

x =-(y-3)^2

y-3 = sqr(-x)

y = + or -sqr(-x) +3

f^-(x) = + or -sqr(-x) +3

It's leftward opening parabola with vertex (0,3) the reverse of the vertex of the original parabola

two other ordered pairs on the inverse relation are (-9,0) and (-9,6)

but that's not a function. To be a function use either the top half or the bottom half of the leftward opening parabola. then you'll need one more ordered pair to fit the inverse function

such as (-4,5) 5=sqr(--4)+3 = 2+3

D. 2x^2 - 1 = 2(1/2)^2 -1 = 2(1/4)-1 = 2/4-1= 1/2-1 = -1/2

that's less than C 2x +2= 2(1/2)+2 = 1 +2 = 3.

-1/2 is not greater than 3

Jody is right. Kayla is wrong

or just let x =0

then 2x^2-1 = -1

and 2x+2 = 2

2>-1

the quadratic is an upward opening parabola with vertex (0,-1)=minimum point = y intercept

the linear equation is a straight line with y intercept =2. 2>-1

2x^2-1 = 2x+1

2x^2-2x -2 =0

x^2-x -1 =0

x = 1/2 + or - (1/2)sqr(1+4)

x = .5 + or - .5sqr5 = about -.6 and 1.6

are two x values where C's value = D's value

for any value between -.6 and +1.6, C's value will be greater than D's value

such as -1/2, 0, 1/2, 1, or 3/2

Answer 2

In the first part of your question, you are to graph the function f(x) = - x² + 6x - 9 and its inverse.

I usually change f(x) to y when I know I'll need to graph the function and when I'm finding the inverse equation. So, y = - x² + 6x - 9 .

Note that the leading coefficient is negative, which means the parabola will open downward. (Specifically, the leading coefficient is -1, though the 1 is not written.) The vertex will be the max of the function (the highest point on the parabola).

The most common ways to find the vertex of a quadratic function (without technology) are:

1. Write the function in vertex form: y = a(x - h)² + k
2. Use the expression -b/2a to find the x-coordinate of the vertex

If the function had been given in vertex form, that would have been the quickest method to find the vertex. Since it was not given in vertex form, we'll use the second method.

Standard form of a quadratic equation is y = ax² + bx + c. (Some prefer ax² + bx + c = 0.)

So, in this problem, a = -1, b = 6, c = -9.

Therefore, -b/2a = -6/-2 = 3

Since the x-coordinate of the vertex is 3, plug in 3 for x in the original function to find the y-coordinate of the vertex.

Once you have the vertex plotted on your graph, you can plug in a few more values for x and solve for y. I suggest you find at least four more points on the graph. You should see that the parabola opens downward.

To find the inverse of this function, you can just switch the x- and y-coordinates of the points you found on the graph of the original function. For example, the point (1, -4) is on the original graph, so (-4, 1) is on the graph of the inverse. The vertex of the inverse is the same point as the vertex of the original function.

Visually, you should see that the inverse of a function is the reflection of the function over the line y = x. Notice that the inverse is NOT a function unless you limit the domain of the original function, which you didn't mention. (The inverse is a "sideways" parabola and does not pass the vertical line test.)

FYI: You could find the actual equation of the inverse by switching x and y in the original function, to make it x = - y² + 6y - 9. You would then have to rewrite it by solving for y, so I wouldn't recommend that method in this problem. You were NOT asked for the EQUATION of the inverse, just the graph and three points on the graph, so you don't need to do this for this problem.

Hope this helps.