This week in math we learned how to convert equations in general form, where you can identify the y intercept, into standard form (or vertex) form. This is an important skill because in order to be able to graph a quadratic function you need the vertex, whether there is a stretch, and if the parabolla is facing upwards or down. I chose this topic because I like how similar it is to our last unit where we learned to complete the square.
Lets look at some examples…
Example A.
Step 1. In my first step I divide my B term AKA the middle term by 2 and square the product of that division. After dividing the term and squaring I am left with 9. To keep the equation in balance I add positive 9 and negative 9.
Step 2. My next step, the negative 9 can leave the brackets and there is no stretch that it needs to be multiplied with so its value remainss the same. Now we can add our two constants of negative 9 and one together leaving us with our terms in the brackets and negative 8 outside the brackets.
Step 3. My last step is to combine the inside terms. To do so; put x and b/2 into brackets and square it, with our constant of -8 remaining on the outside. To find the vertex simply take the value inside of the brackets and change its value which would give us -3 as our x value of the vertex. The Y value is our constant on the outside, -8. It remains negative.
Lets take a look at an uglier example…
Example B.
Step 1. My first step is to get rid of the leading coefficent infront of my A value. To do this out the A and B values in brackets and divide them by 2.
Step 2. My next step is to divide the B term by 2. Since my B term is a fraction over 2, multiply the denominator by 2 giving the new product after being divided. Square this term and add it into the brackets, as well as its negative value to balance out the equation. To get rid of the extra negative value we added, multiply it by the coefficent 2 we factored out so it can leave the brackets.
Step 3. Now we may simplify. Combine X plus the divided b term in brackets and square it. For our constants on the outside of the brackets, we can first divide -50/16 by 2 giving -25/8. Because we are combining -3 with -25/8; the denominators need to be the same so we must multiply -3/1 by 8. This leaves us with our final inside terms and our constant of -49/8. To find the vertex, take the value inside our bracket and change its sign to negative giving us our x value. This means there is a horizontal shift to the left. Our why value is the constant outside the brackets. This means there is a vertical shift downwards. The 2 outside the brackets tells us there is a stretch of 2.
Glossary :
Horizontal shift : inside changes that affect the input (x−) axis values and shift the function left or right.
Vertical shift : The outside changes that affect the output (y−) axis values and shift the function up or down.
Vertex/standard form : y = a ( x − h ) 2 + k
Stretch : (narrowing, widening) of the parabolla