Question
Part 1
I researched logos from various brands, including Nike and Apple. The Nike logo, known as the "Swoosh," is appealing due to its simplicity and dynamic shape, which conveys motion and energy. Apple’s logo, a stylized apple with a bite taken out, is memorable and sophisticated, using clean lines and a modern aesthetic to represent innovation and quality. Both logos effectively utilize negative space and minimalism, making them easily recognizable and versatile across various media.
Part 2
For my logo design, I chose to create a stylized letter "X" using the parent function of a quadratic equation:
. I applied a vertical stretch and a horizontal compression. The transformations produced an elongated "X" that is symmetrical and visually striking. Additionally, I incorporated circular arcs at the tips of the "X" to represent inclusivity and connection.
Part 3
The specifics of my design include the equations:
For the quadratic curve of the "X":
(vertical stretch).
For the horizontal compression:
.
The domain restriction is
, which allows the logo to maintain its intended shape. The colors chosen are a deep navy blue for the main elements, symbolizing trust and reliability, paired with a vibrant orange for the arcs, representing creativity and energy.
My pitch for this logo is that it encapsulates both modernity and stability, appealing to customers who value innovation and quality. The striking color contrast and unique shape ensure it stands out in a crowded market, making it memorable and relatable, ideal for attracting a diverse customer base.
Is the math in this correct?
I researched logos from various brands, including Nike and Apple. The Nike logo, known as the "Swoosh," is appealing due to its simplicity and dynamic shape, which conveys motion and energy. Apple’s logo, a stylized apple with a bite taken out, is memorable and sophisticated, using clean lines and a modern aesthetic to represent innovation and quality. Both logos effectively utilize negative space and minimalism, making them easily recognizable and versatile across various media.
Part 2
For my logo design, I chose to create a stylized letter "X" using the parent function of a quadratic equation:
. I applied a vertical stretch and a horizontal compression. The transformations produced an elongated "X" that is symmetrical and visually striking. Additionally, I incorporated circular arcs at the tips of the "X" to represent inclusivity and connection.
Part 3
The specifics of my design include the equations:
For the quadratic curve of the "X":
(vertical stretch).
For the horizontal compression:
.
The domain restriction is
, which allows the logo to maintain its intended shape. The colors chosen are a deep navy blue for the main elements, symbolizing trust and reliability, paired with a vibrant orange for the arcs, representing creativity and energy.
My pitch for this logo is that it encapsulates both modernity and stability, appealing to customers who value innovation and quality. The striking color contrast and unique shape ensure it stands out in a crowded market, making it memorable and relatable, ideal for attracting a diverse customer base.
Is the math in this correct?
Answers
GPT-4o mini
To evaluate the math in your logo design and transformations, I'll break down the points you mentioned and provide some general guidance on the functions you may be referencing:
1. **Parent Function of a Quadratic Equation**: The parent function typically refers to \( f(x) = x^2 \). If you're creating a stylized "X," it implies you may be using two symmetrical parabolas to form the shape.
2. **Vertical Stretch and Horizontal Compression**:
- A vertical stretch of a quadratic function can be represented as \( f(x) = a(x^2) \), where \( a > 1 \).
- A horizontal compression can be represented by modifying the input to the function, typically shown as \( f(x) = (bx)^2 \), where \( b > 1 \).
3. **Combining Transformations**: If you're combining both in your design, you might be describing a function that looks something like \( y = a(bx)^2 \).
4. **Equations for the "X" Shape**: Generally, if you were forming an "X," you might have two quadratic equations, one for each side of the "X":
- For the left side: \( y = -a(bx)^2 + c \) (opening down).
- For the right side: \( y = a(bx)^2 + c \) (opening up).
- Where \( c \) controls the vertical position.
5. **Domain Restriction**: The domain restriction you mention (which wasn't specified in your text but is noted as critical) would restrict the values of \( x \) to maintain the "X" shape. You might want to define it clearly (e.g., \([-1, 1]\)) based on how wide or narrow you want the "X" to appear.
6. **Circular Arcs**: The arcs at the tips can be represented by circles or part of circles added to the ends of each arm of your "X." The equations for circular arcs could take a form of the standard circle equation, and you would use only parts of it to form arcs.
7. **Color Choices & Symbolism**: While this aspect isn’t mathematical, pairing deep navy blue for trust and vibrant orange for creativity is a good approach in branding and color theory.
Overall, you seem to have the right idea for creating a logo with a mathematical basis. If you want to provide the exact equations or clarify your transformations and calculations, I could help you ensure accuracy in your mathematical design.
1. **Parent Function of a Quadratic Equation**: The parent function typically refers to \( f(x) = x^2 \). If you're creating a stylized "X," it implies you may be using two symmetrical parabolas to form the shape.
2. **Vertical Stretch and Horizontal Compression**:
- A vertical stretch of a quadratic function can be represented as \( f(x) = a(x^2) \), where \( a > 1 \).
- A horizontal compression can be represented by modifying the input to the function, typically shown as \( f(x) = (bx)^2 \), where \( b > 1 \).
3. **Combining Transformations**: If you're combining both in your design, you might be describing a function that looks something like \( y = a(bx)^2 \).
4. **Equations for the "X" Shape**: Generally, if you were forming an "X," you might have two quadratic equations, one for each side of the "X":
- For the left side: \( y = -a(bx)^2 + c \) (opening down).
- For the right side: \( y = a(bx)^2 + c \) (opening up).
- Where \( c \) controls the vertical position.
5. **Domain Restriction**: The domain restriction you mention (which wasn't specified in your text but is noted as critical) would restrict the values of \( x \) to maintain the "X" shape. You might want to define it clearly (e.g., \([-1, 1]\)) based on how wide or narrow you want the "X" to appear.
6. **Circular Arcs**: The arcs at the tips can be represented by circles or part of circles added to the ends of each arm of your "X." The equations for circular arcs could take a form of the standard circle equation, and you would use only parts of it to form arcs.
7. **Color Choices & Symbolism**: While this aspect isn’t mathematical, pairing deep navy blue for trust and vibrant orange for creativity is a good approach in branding and color theory.
Overall, you seem to have the right idea for creating a logo with a mathematical basis. If you want to provide the exact equations or clarify your transformations and calculations, I could help you ensure accuracy in your mathematical design.