Precalc Fin

Give a clear and precise mathematical definition of a function.

A function is a relation in which each input (x-value) corresponds to exactly one output (y-value). This means that for every value in the domain, there is one and only one associated value in the range. A function can be represented using equations, graphs, tables, or mapping diagrams. Graphically, a relation is a function if it passes the vertical line test, meaning no vertical line intersects the graph at more than one point.

Compare and contrast the graphs of the sine and cosine function.

The sine and cosine functions are both periodic functions with a period of 2π, and their graphs have similar shapes, with oscillations between -1 and 1 in their parent forms. However, they differ in their starting points:

• The sine function begins at the origin (0, 0) and follows the pattern: midline → maximum → midline → minimum → midline.

• The cosine function starts at its maximum point (0, 1) and follows the pattern: maximum → midline → minimum → midline → maximum.

Essentially, the cosine graph is a horizontal shift of the sine graph to the left by π/2 radians. Despite this shift, both graphs share the same amplitude, period, and general wave shape.

Give an everyday example in which you have applied or will apply your new mathematical learning in your art area.

As a photographer, I apply mathematical thinking when working with exposure curves and lighting adjustments. Understanding wave patterns—like sine and cosine curves—helps when adjusting highlights, midtones, and shadows in photo editing. I also rely on function transformations when correcting lens distortions, which involve manipulating curves to restore straight lines. Additionally, concepts like symmetry, scale, and proportion all involve mathematical reasoning, showing that math is deeply integrated into artistic composition and design.

Reflect on your learning experience throughout this course and explain how you can use this to inform your performance next year for your new math course.

This course taught me that math is not just about memorizing formulas, but about recognizing patterns, relationships, and strategies for problem-solving. Even when the content was challenging, I learned the importance of persistence, practice, and breaking problems down into manageable steps. Next year, I’ll bring this mindset into my new math course by focusing on understanding the concepts, staying consistent with practice, and asking for help when needed. This course helped me grow more confident in my mathematical abilities, and I now know how to approach future topics with resilience and curiosity.

Given the standard equation, explain how a, b, c, and d individually impact the graph of the cosine function.

In the standard cosine function y = a·cos(bx – c) + d, each parameter transforms the graph in a specific way:

• a (amplitude): This affects the height of the wave. The graph stretches vertically when |a| > 1 and compresses when 0 < |a| < 1. If a is negative, the graph is reflected over the midline.

• b (frequency/period): This affects the horizontal stretch. The period of the function is 2π ÷ |b|, so larger values of b make the graph cycle more quickly (narrower waves), while smaller values make the wave wider.

• c (phase shift): This shifts the graph horizontally. The graph moves right if c is positive and left if c is negative. The amount of the shift is c ÷ b.

• d (vertical shift): This moves the midline of the graph up or down, depending on the sign of d. It changes the vertical position of the wave but not its shape.

These parameters combine to customize the cosine graph to fit various scenarios and applications.