Let's analyze the function you provided:

$f(x) = -0.15(x - 5)^2 + 5.75$

This function is a quadratic function in vertex form, $f(x) = a(x-h)^2 + k$, where:

• $a = -0.15$: This determines the direction and "width" of the parabola.
  • Since $a < 0$, the parabola opens downward.
• $h = 5$: This is the $x$-coordinate of the vertex.
• $k = 5.75$: This is the $y$-coordinate of the vertex.

Key Features of the Function:

1. Vertex: The vertex of the parabola is at $(h, k) = (5, 5.75)$.

2. Axis of Symmetry: The axis of symmetry is the vertical line passing through the vertex: $x = 5$

3. Maximum Value: Since the parabola opens downward, the maximum value of the function occurs at the vertex: $f(5) = 5.75$

4. Intercepts:

   • Y-Intercept: To find the $y$-intercept ($f(0)$), substitute $x = 0$: $f(0) = -0.15(0 - 5)^2 + 5.75 = -0.15(25) + 5.75 = -3.75 + 5.75 = 2$ So, the $y$-intercept is $(0, 2)$.
   • X-Intercepts: Solve $f(x) = 0$: $-0.15(x - 5)^2 + 5.75 = 0$ Rearrange: $(x - 5)^2 = \frac{5.75}{0.15} = 38.33\ldots$ Take the square root: $x - 5 = \pm \sqrt{38.33}$ Approximate: $x - 5 = \pm 6.19$ Solve for $x$: $x = 5 \pm 6.19$ So: $x \approx 11.19 \quad \text{and} \quad x \approx -1.19$ The $x$-intercepts are approximately $(-1.19, 0)$ and $(11.19, 0)$.

5. Domain and Range:

   • Domain: Since this is a quadratic function, the domain is all real numbers: $(-\infty, \infty)$
   • Range: The function has a maximum value at the vertex, so: $(-\infty, 5.75]$

Would you like me to plot the function or go deeper into any of these features?

Follow-Up Questions:

1. Would you like a visual graph of this function?
2. Should we verify the intercepts more precisely?
3. Would you like more details on how the range was determined?
4. Should I explore transformations related to this function?
5. Do you need real-life applications of such functions?

Tip:

When solving for intercepts, always simplify your calculations as much as possible to avoid unnecessary errors!