SOLUTION: Graph f(x) = x^2 - 4x -5

Algebra ->  Graphs -> SOLUTION: Graph f(x) = x^2 - 4x -5      Log On


   

Question 171603: Graph f(x) = x^2 - 4x -5
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
In order to graph f%28x%29=x%5E2-4x-5, we can follow the steps:


Step 1) Find the vertex (the vertex is the either the highest or lowest point on the graph)


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 the points over the axis of symmetry to get two more points on the right side of the parabola (remember a parabola is symmetric).


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


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=x%5E2-4x-5, we can see that a=1, b=-4, and c=-5.


x=%28-%28-4%29%29%2F%282%281%29%29 Plug in a=1 and b=-4.


x=%284%29%2F%282%281%29%29 Negate -4 to get 4.


x=%284%29%2F%282%29 Multiply 2 and 1 to get 2.


x=2 Divide.


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


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


f%28x%29=x%5E2-4x-5 Start with the given function.


f%282%29=%282%29%5E2-4%282%29-5 Plug in x=2.


f%282%29=4-4%282%29-5 Square 2 to get 4.


f%282%29=4-8-5 Multiply -4 and 2 to get -8.


f%282%29=-9 Combine like terms.


So the y-coordinate of the vertex is y=-9.


So the vertex is .

------------------------------------------------------------------------------------


Step 2)

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




Since the x coordinate of the vertex is 2, this means that the x values 0 and 1 are to the left of the vertex. So to find the points on the graph that are to the left of the vertex, simply plug in x=0 and x=1 to find their corresponding y coordinates

Finding the first point:

f%28x%29=x%5E2-4x-5 Start with the given function


f%280%29=%280%29%5E2-4%280%29-5 Plug in x=0


f%280%29=0-4%280%29-5 Square 0 to get 0


f%280%29=0-0-5 Multiply 4 by 0 to get 0


f%280%29=5 Combine like terms


So the 1st point on the left side of the axis of symmetry is (0,-5)


--------------------------------------------------------

Finding the second point:


f%28x%29=x%5E2-4x-5 Start with the given function


f%281%29=%281%29%5E2-4%281%29-5 Plug in x=1


f%281%29=1-4%281%29-5 Square 1 to get 1


f%281%29=1-4-5 Multiply 4 by 1 to get 4


f%281%29=-8 Now combine like terms


So the 2nd point on the left side is (1,-8)


------------------------------------------------------------------------------------


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=2)


This means the y-value for x=1 is equal to the y-value of x=3. So when x=3, y=-8 which gives us the point (3,-8). So we essentially reflected the point (1,-8) over to (3,-8).


Also, the y-value for x=0 is equal to the y-value of x=4. So when x=4, y=-5 which gives us the point (4,-5). So we essentially reflected the point (0,-5) over to (4,-5).


------------------------------------------------------------------------------------


Step 4)

Plotting the points:



Now lets make a table of the values we have calculated 



 xy


 0-5
 

 1-8
 

 2-9
 

 3-8
 

 4-5



Now let's plot the points:




------------------------------------------------------------------------------------


Step 5)

Drawing a curve through all of the points:




Now draw a curve through all of the points to graph y=x%5E2-4x-5:


Graph of y=x%5E2-4x-5