document.write( "Question 171579: Can you please show me how to graph each quadratic equation using its properties, and also based on the graph, determine the domain and range of the quadratic function.\r
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Algebra.Com's Answer #126755 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
In order to graph $\"f%28x%29=-x%5E2%2B2x%2B8\"$ , 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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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 $\"y=-x%5E2%2B2x%2B8\"$ , we can see that $\"a=-1\"$ , $\"b=2\"$ , and $\"c=8\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"x=%28-%282%29%29%2F%282%28-1%29%29\"$ Plug in $\"a=-1\"$ and $\"b=2\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"x=%28-2%29%2F%28-2%29\"$ Multiply 2 and $\"-1\"$ to get $\"-2\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"x=1\"$ Divide.\r
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\n" ); document.write( "\n" ); document.write( "So the x-coordinate of the vertex is $\"x=1\"$ . Note: this means that the axis of symmetry is also $\"x=1\"$ .\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( " $\"y=-x%5E2%2B2x%2B8\"$ Start with the given equation.\r
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\n" ); document.write( "\n" ); document.write( " $\"y=-%281%29%5E2%2B2%281%29%2B8\"$ Plug in $\"x=1\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"y=-1%281%29%2B2%281%29%2B8\"$ Square $\"1\"$ to get $\"1\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"y=-1%2B2%281%29%2B8\"$ Multiply $\"-1\"$ and $\"1\"$ to get $\"-1\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"y=-1%2B2%2B8\"$ Multiply $\"2\"$ and $\"1\"$ to get $\"2\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"y=9\"$ 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

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

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\n" ); document.write( "\n" ); document.write( "Lets find the value of y when x=-1\r
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\n" ); document.write( "\n" ); document.write( " $\"f%28x%29=-x%5E2%2B2x%2B8\"$ Start with the given polynomial\r
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\n" ); document.write( "\n" ); document.write( " $\"f%28-1%29=-%28-1%29%5E2%2B2%28-1%29%2B8\"$ Plug in $\"x=-1\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"f%28-1%29=-%281%29%2B2%28-1%29%2B8\"$ Square 1 to get 1\r
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\n" ); document.write( "\n" ); document.write( " $\"f%28-1%29=-%281%29-2%2B8\"$ Multiply 2 by -1 to get -2\r
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\n" ); document.write( "\n" ); document.write( " $\"f%28-1%29=5\"$ Now combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So our 1st point is (-1,5)\r
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\n" ); document.write( "\n" ); document.write( "----Now lets find another point----\r
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\n" ); document.write( "\n" ); document.write( "Lets find the y value when x=0\r
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\n" ); document.write( "\n" ); document.write( " $\"f%28x%29=-x%5E2%2B2x%2B8\"$ Start with the given polynomial\r
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\n" ); document.write( "\n" ); document.write( " $\"f%280%29=-%280%29%5E2%2B2%280%29%2B8\"$ Plug in $\"x=0\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"f%280%29=0%2B2%280%29%2B8\"$ Square 0 to get 0\r
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\n" ); document.write( "\n" ); document.write( " $\"f%280%29=0%2B0%2B8\"$ Multiply 2 by 0 to get 0\r
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\n" ); document.write( "\n" ); document.write( " $\"f%280%29=8\"$ Combine terms\r
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\n" ); document.write( "\n" ); document.write( "So our 2nd point is (0,8)\r
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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 $\"x=1\"$ )\r
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\n" ); document.write( "\n" ); document.write( "This means the y-value for $\"x=0\"$ is equal to the y-value of $\"x=2\"$ . So when $\"x=2\"$ , $\"y=8\"$ . So we've reflected the point (0,8) over to (2,8)\r
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\n" ); document.write( "\n" ); document.write( "Also, the y-value for $\"x=-1\"$ is equal to the y-value of $\"x=3\"$ . So when $\"x=3\"$ , $\"y=5\"$ . So we've reflected the point (-1,5) over to (3,5)\r
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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( " x y
 -1 5
 0 8
 1 9
 2 8
 3 5\r\n" );
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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 $\"y=-x%5E2%2B2x%2B8\"$ :\r
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Graph of $\"y=-x%5E2%2B2x%2B8\"$ \r
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\n" ); document.write( "\n" ); document.write( "From the graph, we can determine the following:\r
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\n" ); document.write( "\n" ); document.write( "Range: $\"y%3C=9\"$ \n" ); document.write( "

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