document.write( "Question 171603: Graph f(x) = x^2 - 4x -5 \n" ); document.write( "
Algebra.Com's Answer #126753 by jim_thompson5910(35256)\"\" \"About 
You can put this solution on YOUR website!
In order to graph \"f%28x%29=x%5E2-4x-5\", we can follow the steps:\r
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\n" ); document.write( "\n" ); document.write( "Step 1) Find the vertex (the vertex is the either the highest or lowest point on the graph)\r
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\n" ); document.write( "\n" ); document.write( "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)\r
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\n" ); document.write( "\n" ); document.write( "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).\r
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\n" ); document.write( "\n" ); document.write( "Step 4) Plot all of the points found (including the vertex)\r
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\n" ); document.write( "\n" ); document.write( "Step 5) Draw a curve through all of the points to graph the parabola\r
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\n" ); document.write( "\n" ); document.write( "Let's go through these steps in detail\r
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\n" ); document.write( "\n" ); document.write( "Step 1)

Finding the vertex:

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\n" ); document.write( "\n" ); document.write( "In order to find the vertex, we first need to find the x-coordinate of the vertex.\r
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\n" ); document.write( "\n" ); document.write( "To find the x-coordinate of the vertex, use this formula: \"x=%28-b%29%2F%282a%29\".\r
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\n" ); document.write( "\n" ); document.write( "\"x=%28-b%29%2F%282a%29\" Start with the given formula.\r
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\n" ); document.write( "\n" ); document.write( "From \"y=x%5E2-4x-5\", we can see that \"a=1\", \"b=-4\", and \"c=-5\".\r
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\n" ); document.write( "\n" ); document.write( "\"x=%28-%28-4%29%29%2F%282%281%29%29\" Plug in \"a=1\" and \"b=-4\".\r
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\n" ); document.write( "\n" ); document.write( "\"x=%284%29%2F%282%281%29%29\" Negate \"-4\" to get \"4\".\r
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\n" ); document.write( "\n" ); document.write( "\"x=%284%29%2F%282%29\" Multiply 2 and \"1\" to get \"2\".\r
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\n" ); document.write( "\n" ); document.write( "\"x=2\" Divide.\r
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\n" ); document.write( "\n" ); document.write( "So the x-coordinate of the vertex is \"x=2\". Note: this means that the axis of symmetry is also \"x=2\".\r
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\n" ); document.write( "\n" ); document.write( "Now that we know the x-coordinate of the vertex, we can use it to find the y-coordinate of the vertex.\r
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\n" ); document.write( "\n" ); document.write( "\"f%28x%29=x%5E2-4x-5\" Start with the given function.\r
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\n" ); document.write( "\n" ); document.write( "\"f%282%29=%282%29%5E2-4%282%29-5\" Plug in \"x=2\".\r
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\n" ); document.write( "\n" ); document.write( "\"f%282%29=4-4%282%29-5\" Square \"2\" to get \"4\".\r
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\n" ); document.write( "\n" ); document.write( "\"f%282%29=4-8-5\" Multiply \"-4\" and \"2\" to get \"-8\".\r
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\n" ); document.write( "\n" ); document.write( "\"f%282%29=-9\" Combine like terms.\r
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\n" ); document.write( "\n" ); document.write( "So the y-coordinate of the vertex is \"y=-9\".\r
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\n" ); document.write( "\n" ); document.write( "So the vertex is .\r
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\n" ); document.write( "\n" ); document.write( "Step 2)

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

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\n" ); document.write( "\n" ); document.write( "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\r
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\n" ); document.write( "\n" ); document.write( "Finding the first point:\r
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\n" ); document.write( "\n" ); document.write( "\"f%28x%29=x%5E2-4x-5\" Start with the given function\r
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\n" ); document.write( "\n" ); document.write( "\"f%280%29=%280%29%5E2-4%280%29-5\" Plug in \"x=0\"\r
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\n" ); document.write( "\n" ); document.write( "\"f%280%29=0-4%280%29-5\" Square 0 to get 0\r
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\n" ); document.write( "\n" ); document.write( "\"f%280%29=0-0-5\" Multiply 4 by 0 to get 0\r
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\n" ); document.write( "\n" ); document.write( "\"f%280%29=5\" Combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So the 1st point on the left side of the axis of symmetry is (0,-5)\r
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\n" ); document.write( "\n" ); document.write( "Finding the second point:\r
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\n" ); document.write( "\n" ); document.write( "\"f%28x%29=x%5E2-4x-5\" Start with the given function\r
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\n" ); document.write( "\n" ); document.write( "\"f%281%29=%281%29%5E2-4%281%29-5\" Plug in \"x=1\"\r
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\n" ); document.write( "\n" ); document.write( "\"f%281%29=1-4%281%29-5\" Square 1 to get 1\r
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\n" ); document.write( "\n" ); document.write( "\"f%281%29=1-4-5\" Multiply 4 by 1 to get 4\r
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\n" ); document.write( "\n" ); document.write( "\"f%281%29=-8\" Now combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So the 2nd point on the left side is (1,-8)\r
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\n" ); document.write( "\n" ); document.write( "Step 3)

Reflecting the two points over the axis of symmetry:

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\n" ); document.write( "\n" ); document.write( "Now remember, the parabola is symmetrical about the axis of symmetry (which is \"x=2\")\r
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\n" ); document.write( "\n" ); document.write( "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).\r
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\n" ); document.write( "\n" ); document.write( "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).\r
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\n" ); document.write( "\n" ); document.write( "Step 4)

Plotting the points:

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\n" ); document.write( "\n" ); document.write( "Now lets make a table of the values we have calculated\r
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document.write( "
 xy
 0-5
 1-8
 2-9
 3-8
 4-5
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\n" ); document.write( "\n" ); document.write( "Now let's plot the points:\r
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Drawing a curve through all of the points:

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\n" ); document.write( "\n" ); document.write( "Now draw a curve through all of the points to graph \"y=x%5E2-4x-5\":\r
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\n" ); document.write( "\n" ); document.write( " Graph of \"y=x%5E2-4x-5\"
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