Parent Function -> y = x²
My Function -> y = ⅓(x+1)² - 3
The Value [hehe] of a, h and k
a is the coefficient of the binomial in which x resides, so modifying this value grants great control over the final value of x. Making the value of a greater than 0 but less than 1 makes the y value always lower than if the identical x value was plugged into the parent function. In graphs that open upwards
[a > 0], having a y value consistently lower than what would result from the parent function given the same value of x produces a graph which is “flatter” – where achieving the same y value as a point on the parent function requires an x value that is further from 0.
I’ve nullified [made equal to 0] both h and k on the function I was given to make comparing both values of a easier.
If the value of a is less than 0, the graph reflects vertically and is considered “opening down”. Much of the same rules apply here. Values of a greater than -1 and lesser than 0 produce a flatter graph [see picture above] but flipped vertically [the vertex is unaffected] and values lesser than -1 produce a thinner graph.
h is the value inside of the brackets and is responsible for shifting the parabola horizontally. Note that it is represented as (x – h)², meaning that if you want the parabola to be shifted to the right [in the positive direction], the value of h has to be negative to produce a double-negative [positive] number in the final equation. Thus – positive values of h make the parabola shifted to the left and negative h values shift it to the right.
Finally, k is the lone constant sitting at the end of the function and shifts the graph vertically. It’s not as confusing to compute as h – positive k values shift the graph vertically and the inverse is true for negative values. Note that adjusting both h and k affect the position of the vertex.
Core Competency Questions
I’ve represented the same idea numerous ways when I both said and shown what changing the a value would do via written words and embedded images and expressed these ideas using relevant mathematical vocabulary like “opening down”, “greater than / less than”, “shifting”, “double-negative” and “constant” which makes my explanation less dependant on guesswork to interpret. Additionally, I’ve organized the sentences about a, h and k into their own separate paragraphs and always started those paragraphs with the corresponding letter that was talked about and separated those paragraphs with spacers.