The function given is $f(x) = 2x - x^2 + 8$, which represents a quadratic function. Let's go through the process of finding the axis of symmetry, domain, and range, as well as sketching the graph.

Step 1: Rewrite the function in standard form

The given function is: $f(x) = -x^2 + 2x + 8$ This is a quadratic function in the form $ax^2 + bx + c$, where $a = -1$, $b = 2$, and $c = 8$.

Step 2: Find the axis of symmetry

The formula for the axis of symmetry of a quadratic function is: $x = -\frac{b}{2a}$ Substitute the values of $a = -1$ and $b = 2$: $x = -\frac{2}{2(-1)} = 1$ So, the axis of symmetry is $x = 1$.

Step 3: Find the vertex

The x-coordinate of the vertex is the same as the axis of symmetry, $x = 1$. To find the y-coordinate, substitute $x = 1$ back into the function: $f(1) = -(1)^2 + 2(1) + 8 = -1 + 2 + 8 = 9$ So, the vertex is $(1, 9)$.

Step 4: Find the domain

Since this is a quadratic function, the domain is all real numbers: $\text{Domain: } (-\infty, \infty)$

Step 5: Find the range

The coefficient of $x^2$ is negative ($a = -1$), so the parabola opens downward. The maximum value occurs at the vertex, $(1, 9)$. Therefore, the range is: $\text{Range: } (-\infty, 9]$

Summary of Results:

• Axis of symmetry: $x = 1$
• Domain: $(-\infty, \infty)$
• Range: $(-\infty, 9]$

Would you like me to explain any of these steps in more detail?

Here are 5 questions for further exploration:

1. How would the graph change if the sign of the $x^2$ term was positive?
2. What is the significance of the axis of symmetry in quadratic functions?
3. Can you find the x-intercepts of this function by factoring or using the quadratic formula?
4. What are the key differences between the vertex form and standard form of a quadratic equation?
5. How can you determine if a quadratic function has a maximum or minimum value?

Tip: The axis of symmetry always passes through the vertex of a parabola. Use this fact to quickly find one key feature of the graph!