Equations and Colors
Explanation of a, h, and k in Vertex Form
In my graph, the black parabola represents the parent function y=x2, with its vertex at (0,0). Because a=1, h=0, and k=0, there’s no vertical or horizontal shift, and it has a standard width. This function serves as a reference for comparing the effects of changing a, h, and k on other parabolas.
In the green parabola, represented by y=(x+6)2+2, the vertex shifts to (−6,2) because of the values of h and k. The h=−6 moves the vertex left by 6 units, while k=2 shifts it up by 2 units, resulting in a new vertex position. This parabola opens upward since a=1 is positive, and it has the same standard width as the parent function because the magnitude of a remains 1.
The purple parabola, y=(x+6)2+3, is similar to the green parabola in terms of horizontal shift, but its vertex is slightly higher at (−6,3). This is because the k value is 3, which shifts the vertex up by an additional unit compared to the green parabola. This shows how changes in k adjust the vertical position of the parabola while keeping other characteristics like width and direction the same.
The orange parabola is defined by y=−0.5(x−3)2+2, where a=−0.5, h=3, and k=2. Here, h=3 shifts the vertex right by 3 units, and k=2 moves it up by 2 units, placing the vertex at (3,2). The negative a-value flips the parabola to open downwards, and since ∣a∣ is 0.5 (less than 1), the parabola is wider than the parent function due to a horizontal stretch. The combination of a negative a-value and a wide opening creates a distinctly downward-facing, wide shape compared to other parabolas on the graph.
In the blue parabola y=−0.25(x−3)2+1, the parameters h=3 and k=1 position the vertex at (3,1). The negative a-value of −0.25 causes the parabola to open downward, while the small magnitude of (a) makes the parabola even wider than the orange parabola. This shows that as ∣a∣ decreases, the parabola stretches horizontally, creating a broader curve.
Lastly, the red parabola y=0.5(x−7)2 demonstrates the effect of a positive a-value with h=7 and k=0. The h=7 shifts the vertex right to (7,0), and with k=0, there is no vertical shift. Since a=0.5, the parabola opens upwards but is wider than the parent function, showing that ∣a∣ less than 1 makes the parabola broader. The positive a-value keeps it facing upward, contrasting with the downward-facing blue and orange parabolas.
Self-Assessment
1)How did you represent the same mathematical idea in multiple ways in this assignment? I represented the transformation of quadratic equations through both visual graphs and algebraic equations. By graphing each equation, I could visually demonstrate how changing a, h, and k affected each parabola’s shape and position.
2)
State some of the relevant mathematical vocabulary words you used to demonstrate your understanding and their definitions.
a) Vertex: The highest or lowest point on a parabola, depending on its direction.
b) Axis of Symmetry: A vertical line that passes through the vertex, dividing the parabola into two mirror images.
c) Direction: Refers to whether the parabola opens upwards or downwards. I found these definitions in my class notes.
3) How did you use formatting to share your information in a clear and organized way? I used headings and bullet points to separate each section clearly. This helped organize the equations, explanations, and self-assessment in a way that is easy to read and follow.