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Forms, Forms and Forms What forms of quadratic equations have we learnt about so far? What forms of quadratic equations have we learnt about so far? Standard Form y=ax 2 +bx+c Standard Form y=ax 2 +bx+c Factored Form y=a(x-r)(x-s) Factored Form y=a(x-r)(x-s)

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What useful info? Remember, Standard form tells us some useful things: Remember, Standard form tells us some useful things: 1) Direction of opening 1) Direction of opening 2) y intercept 2) y intercept

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What useful info? Remember, factored form also tells us some useful info Remember, factored form also tells us some useful info 1) Direction of opening 1) Direction of opening 2) X intercepts 2) X intercepts

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Now there is another form to make your life more convenient

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It is called Vertex Form! Here it is: Here it is: y=a(x-h) 2 + k y=a(x-h) 2 + k

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What does it all mean? y=a(x-h) 2 +k y=a(x-h) 2 +k Direction of opening X coordinate of vertex. Watch the sign Y coordinate of vertex

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Can you see why it might be useful? Vertex form tells us Vertex form tells us 1) Direction of opening 1) Direction of opening 2) Coordinates of the Vertex 2) Coordinates of the Vertex

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Example Given y=-2(x-4) 2 +5 Given y=-2(x-4) 2 +5 What is the direction of opening? What is the direction of opening? Down Down What are the coordinates of the vertex? What are the coordinates of the vertex? (4,5) (4,5)

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Why is it (+4) Because if the vertex is at (4,5) the (-) sign in front of the "h" will make (+4) show up as (-4) Because if the vertex is at (4,5) the (-) sign in front of the "h" will make (+4) show up as (-4)

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One more example Given y=0.5(x+3) 2 -8 Given y=0.5(x+3) 2 -8 What are the coordinates of the vertex? What are the coordinates of the vertex? (-3,-8) (-3,-8)

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Relating the Standard and Vertex Forms: Completing the Square Unfortunately…..there are some things in life that you just have to remember…..

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Step 1 Insert brackets around the first two terms Insert brackets around the first two terms

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Step 2 Factor out any value in front of the x 2 Factor out any value in front of the x 2 In this case there is nothing to factor out In this case there is nothing to factor out

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Step 3 Take Middle term divide by 2 & square it Take Middle term divide by 2 & square it

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Step 4 Rewrite the equation with the result from step 3 added and subtracted inside the brackets Rewrite the equation with the result from step 3 added and subtracted inside the brackets

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Step 5 Bring the (-) term outside of the bracket remember to remultiply if necessary Bring the (-) term outside of the bracket remember to remultiply if necessary

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Step 6 Combine the two constant terms outside the bracket Combine the two constant terms outside the bracket

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Step 7 Factor trinomial inside the bracket Factor trinomial inside the bracket

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Step 8 Express answer in vertex form Express answer in vertex form

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Step 1 Insert brackets around the first two terms Insert brackets around the first two terms Y =2x 2 + 36x + 170 Y = (2x 2 + 36x) + 170

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Step 2 Factor out any value in front of the x 2 Factor out any value in front of the x 2 In this case there is something to factor out In this case there is something to factor out Y = 2(x 2 + 18x) + 170

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Step 3 Take Middle term divide by 2 & square it Take Middle term divide by 2 & square it

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Step 4 Rewrite the equation with the result from step 3 added and subtracted inside the brackets Rewrite the equation with the result from step 3 added and subtracted inside the brackets Y = 2(x 2 + 18x + 81 – 81) + 170

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Step 5 Bring the (-) term outside of the bracket remember to multiply if necessary. This time it is!!!! Bring the (-) term outside of the bracket remember to multiply if necessary. This time it is!!!! Y = 2(x 2 + 18x + 81) – 162 + 170

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Step 6 Combine the two constant terms outside the bracket Combine the two constant terms outside the bracket Y = 2(x 2 + 18x + 81) + 8

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Step 7 Factor trinomial inside the bracket Factor trinomial inside the bracket Y = 2 (x + 9) ( x + 9) + 8

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Step 8 Express answer in vertex form Express answer in vertex form Y = 2(x + 9) 2 + 8

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