The problem involves analyzing a quadratic function and its transformation compared to the parent function $f(x) = x^2$. Let's break it down based on what is provided and solve the missing details:

Given function: $f(x) = -\frac{1}{2}(x - 5)^2 - 3$

Parent Function:

The parent function is $f(x) = x^2$.

Transformation:

The function $f(x) = -\frac{1}{2}(x - 5)^2 - 3$ has undergone the following transformations compared to $f(x) = x^2$:

• Right by 5 units: $(x - 5)$ indicates a shift to the right by 5 units.
• Down by 3 units: The constant $-3$ shifts the graph down by 3 units.
• Vertical compression and reflection: The factor $-\frac{1}{2}$ compresses the graph vertically (it becomes "wider") and reflects it across the x-axis (flips the parabola).

Domain and Range:

• Domain: Since this is a quadratic function, its domain is all real numbers, $(-\infty, \infty)$.
• Range: The range is limited by the maximum value the function can take, which is $y = -3$ (as the parabola opens downward). Thus, the range is $(-\infty, -3]$.

Intercepts:

• x-intercept(s): To find the x-intercept, set $f(x) = 0$: $0 = -\frac{1}{2}(x - 5)^2 - 3$ $\frac{1}{2}(x - 5)^2 = -3$ No real solutions exist since the square of a real number cannot be negative. Hence, there are no x-intercepts (this is indicated as "none").

• y-intercept: Set $x = 0$ to find the y-intercept: $f(0) = -\frac{1}{2}(0 - 5)^2 - 3 = -\frac{1}{2}(25) - 3 = -\frac{25}{2} - 3 = -\frac{25}{2} - \frac{6}{2} = -\frac{31}{2}$ So, the y-intercept is $\left( 0, -\frac{31}{2} \right)$, which is approximately $(0, -15.5)$.

Extrema (Maximum Point):

The vertex of the parabola is at $(5, -3)$. Since the parabola opens downwards, this point is a maximum.

Increasing/Decreasing Intervals:

• The function decreases on the interval $(-\infty, 5)$ and increases on $(5, \infty)$, as the vertex is at $x = 5$ and the parabola opens downward.

End Behavior:

• As $x \to \infty$, $f(x) \to -\infty$.
• As $x \to -\infty$, $f(x) \to -\infty$ as well.

Would you like further explanation or have any questions?

Here are 5 related questions:

1. How do you find the vertex of a quadratic function in standard form?
2. What effect does a negative coefficient have on a quadratic function?
3. How do you determine if a quadratic function has real x-intercepts?
4. What are the general steps to find the domain and range of a function?
5. How does vertical compression/stretching affect the shape of a quadratic graph?

Tip: To easily identify transformations of a quadratic function, look at the changes made to the variables within and outside the squared term.