SOLUTION: To Graph f(x)> 2x squared - 2x - 1. I really dpnt get how to graph it if you can help that would be fantastic.

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: To Graph f(x)> 2x squared - 2x - 1. I really dpnt get how to graph it if you can help that would be fantastic.      Log On


   

Question 171835: To Graph f(x)> 2x squared - 2x - 1. I really dpnt get how to graph it if you can help that would be fantastic.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!


Table of Contents:
Step 1: Finding the Vertex
Step 2: Finding two points to left of axis of symmetry
Step 3: Reflecting two points to get points right of axis of symmetry
Step 4: Plotting the Points (with table)
Step 5: Graphing the Parabola
Step 6: Graphing the Inequality (by shading the correct region)

In order to graph f%28x%29%3E2x%5E2-2x-1, we need to graph f%28x%29=2x%5E2-2x-1 first



In order to graph f%28x%29=2x%5E2-2x-1, we can follow the steps:


Step 1) Find the vertex (the vertex is the either the highest or lowest point on the graph). Also, the vertex is at the axis of symmetry of the parabola (ie it divides it in two).


Step 2) Once you have the vertex, find two points on the left side of the axis of symmetry (the line that vertically runs through the vertex).


Step 3) Reflect those two points over the axis of symmetry to get two more points on the right side of the axis of symmetry.


Step 4) Plot all of the points found (including the vertex).


Step 5) Draw a curve through all of the points to graph the parabola.


Let's go through these steps in detail


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Step 1)

Finding the vertex:




In order to find the vertex, we first need to find the x-coordinate of the vertex.


To find the x-coordinate of the vertex, use this formula: x=%28-b%29%2F%282a%29.


x=%28-b%29%2F%282a%29 Start with the given formula.


From y=2x%5E2-2x-1, we can see that a=2, b=-2, and c=-1.


x=%28-%28-2%29%29%2F%282%282%29%29 Plug in a=2 and b=-2.


x=%282%29%2F%282%282%29%29 Negate -2 to get 2.


x=%282%29%2F%284%29 Multiply 2 and 2 to get 4.


x=1%2F2 Reduce.


So the x-coordinate of the vertex is x=1%2F2. Note: this means that the axis of symmetry is also x=1%2F2.


Now that we know the x-coordinate of the vertex, we can use it to find the y-coordinate of the vertex.


y=2x%5E2-2x-1 Start with the given equation.


y=2%281%2F2%29%5E2-2%281%2F2%29-1 Plug in x=1%2F2.


y=2%281%2F4%29-2%281%2F2%29-1 Square 1%2F2 to get 1%2F4.


y=1%2F2-2%281%2F2%29-1 Multiply 2 and 1%2F4 to get 1%2F2.


y=1%2F2-1-1 Multiply -2 and 1%2F2 to get -1.


y=-3%2F2 Combine like terms.


So the y-coordinate of the vertex is y=-3%2F2.


So the vertex is .


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Step 2)

Find two points to the left of the axis of symmetry:




Let's find the y value when x=-1


y=2x%5E2-2x-1 Start with the given equation.


y=2%28-1%29%5E2-2%28-1%29-1 Plug in x=-1.


y=2%281%29-2%28-1%29-1 Square -1 to get 1.


y=2-2%28-1%29-1 Multiply 2 and 1 to get 2.


y=2%2B2-1 Multiply -2 and -1 to get 2.


y=3 Combine like terms.


So the first point to the left of the axis of symmetry is (-1,3)


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Let's find the y value when x=0


y=2x%5E2-2x-1 Start with the given equation.


y=2%280%29%5E2-2%280%29-1 Plug in x=0.


y=2%280%29-2%280%29-1 Square 0 to get 0.


y=0-2%280%29-1 Multiply 2 and 0 to get 0.


y=0%2B0-1 Multiply -2 and 0 to get 0.


y=-1 Combine like terms.


So the second point to the left of the axis of symmetry is (0,-1)


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Step 3)

Reflecting the two points over the axis of symmetry:




Now remember, the parabola is symmetrical about the axis of symmetry (which is x=1%2F2)


This means the y-value for x=0 (which is half a unit from the axis of symmetry) is equal to the y-value of x=1 (which is also half a unit from the axis of symmetry). So when x=1, y=-1 which gives us the point (1,-1). So we essentially reflected the point (0,-1) over to (1,-1).


Also, the y-value for x=-1 (which is one and a half units from the axis of symmetry) is equal to the y-value of x=2 (which is also one and a half units from the axis of symmetry). So when x=2, y=3 which gives us the point (2,3). So we essentially reflected the point (-1,3) over to (2,3).


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Step 4)

Plotting the points:




Now lets make a table of the values we have calculated:


xy
-13
0-1
1/2-3/2
1-1
23



Now let's plot the points:



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Step 5)

Drawing a curve through all of the points:




Now draw a curve through all of the points to graph y=2x%5E2-2x-1:


Graph of y=2x%5E2-2x-1



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Step 6)

Graphing the Inequality:




Now because f%28x%29 is the same as y, this means that f%28x%29%3E2x%5E2-2x-1 is identical to y%3E2x%5E2-2x-1.


Now what y%3E2x%5E2-2x-1 tells us is that every point in the shaded region will be above the curve of 2x%5E2-2x-1 (since "y" is greater than the expression)


So f%28x%29%3E2x%5E2-2x-1 looks like:


Photobucket - Video and Image Hosting


Graph of f%28x%29%3E2x%5E2-2x-1 where the boundary is the equation f%28x%29=2x%5E2-2x-1 (note: this should be a dotted/dashed line) and the shaded region in green.


The reason why the line should be dotted is because we are NOT including the boundary (since the sign is a greater than sign)