y = (x^4 - 7) / (4^4).
To analyze the function \( y = \frac{x^4 - 7}{4^4} \), we can simplify it first:
Since \( 4^4 = 256 \), the equation can be rewritten as:
\[ y = \frac{x^4 - 7}{256} \]
This is a polynomial function where the numerator \( x^4 - 7 \) is a quartic polynomial, and the denominator is a constant.
Key Characteristics
Domain: The domain of \( y \) is all real numbers \( x \) since there are no restrictions (the polynomial can take any real value).
Intercepts:
y-intercept: Set \( x = 0 \): \[ y = \frac{0^4 - 7}{256} = \frac{-7}{256} \] So the y-intercept is at \( (0, -\frac{7}{256}) \).
x-intercepts: Set \( y = 0 \): \[ \frac{x^4 - 7}{256} = 0 \implies x^4 - 7 = 0 \implies x^4 = 7 \implies x = \pm \sqrt[4]{7} \] Thus, the x-intercepts are at \( (\sqrt[4]{7}, 0) \) and \( (-\sqrt[4]{7}, 0) \).
Behavior: As \( x \) approaches \( \pm \infty \), \( x^4 \) dominates, thus \( y \to \infty \). As \( x \) approaches \( 0 \), \( y \) approaches \( -\frac{7}{256} \).
Symmetry: The function is even since \( y(x) = y(-x) \). This means the graph is symmetric with respect to the y-axis.
Critical Points and Extrema: To find the critical points, we can take the derivative and set it to zero:
- Using the quotient rule: \[ y' = \frac{(4x^3)(256) - (1)(x^4 - 7)(0)}{256^2} = \frac{1024 x^3}{256} = \frac{4x^3}{1} \] Setting \( y' = 0 \): \[ 4x^3 = 0 \implies x = 0 \] To determine if this is a maximum or minimum, we can examine the second derivative or use the first derivative test.
Second Derivative: \( y'' = 12x^2 \). Since the second derivative is non-negative for all \( x \), \( x = 0 \) is a local minimum.
Summary
- Function: \( y = \frac{x^4 - 7}{256} \)
- Domain: All real numbers (\( \mathbb{R} \))
- Intercepts: y-intercept at \( (0, -\frac{7}{256}) \), x-intercepts at \( (\sqrt[4]{7}, 0) \) and \( (-\sqrt[4]{7}, 0) \)
- Local Minimum: at \( (0, -\frac{7}{256}) \)
You can further analyze the graph based on this information or explore any additional specific aspects you're interested in!
