

The significance of a, h, and k in each parabola–
The blue-dotted line parabola represents the parent function. A parabola at it’s simplest equation.
“f(x) = x^2”, this formulas h, and k values are 0, this can be seen as the parabola is not shifted left or right, or up and down. The a value, being 1 does not change the width of the parabola, leaving it normal.
The red parabola “f(x) = -2(x+4)^2” represents multiple transformations, the opening direction of the parabola, a shift of the vertex, and width of the parabola. Because the a has a negative value (a=-2) the parabola opens downward, as whatever was just squared in the brackets, will become negative when multiplied with the a. For example, if I plugged in x = -6, you will end up with f(x) = -8, a point in the negative y-direction. Furthermore, this parabola has a shift in the x-axis, the value of h = -4, moving the parabola to the negative x direction, this changes the axis-of-symmetry for the parabola. If the absolute value of a is greater than 1, the parabola will be narrow, the absolute value of the a in this equation is greater than 1, causing a more narrow parabola.
The orange parabola “f(x) = (x+6)^2 + 3” demonstrates a transformation in the shift of the parabola in the y-direction. The k value determines the shift of the parabola in the y-direction. Because k=3, the vertex of the parabola is at a point of (-6,3). This k value, will shift every y value up by 3 points.
The purple parabola “f(x) = 1/2(x-8)^2-5” shows multiple transformations, a parabola shifted 8 point in the positive x direction , a parabola shifted -5 points below the x-axis, and a difference in the width of the parabola. The parabola has a h value of 8, which changes the axis-of-symmetry, additionally, the parabola is shifted down below the x axis, which changes the x-intercepts and y-intercepts of the parabola. The absolute value of the a in this parabola is less than 1, but greater than 0, this causes the parabola to be wider than a parent function parabola.
The green parabola “f(x) = -6(x-3)^2 + 3” has transformations that all have already been covered. But this parabola show us one that is shifted up in the y-axis with a “+ k value”, but has a “-a value”, opening downwards and still having two x-intercepts.
- I represented the same mathematical ideas in multiple ways by using many different examples, and explaining how different values effected the parabola. Firstly, in each section of the paragraph I explained what made the parabola differ from the parent function, such as how the k value shifts the parabola depending on its value and being positive or negative.
- Some relevant math vocabulary I used was the parent function, x-intercepts, y-intercepts, and absolute value which all determine the parabolas equation. Using proper vocabulary is a necessity to understanding what each variable in a parabola is responsible for.
- I formatted my graph to display accurate, easy-to-decipher equations and parabolas. By setting the step value in the x and y direction to 1, the scale of the parabola is easier to follow. I exported the image for the graph to avoid mistakes when screenshotting.