The image contains two graphs of quadratic functions and some related questions that help you analyze the properties of the graphs.

Key Points:

1. Quadratic Functions:

   • Quadratic functions are of the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.
   • The graph of a quadratic function is a parabola, which can either open upward or downward depending on the sign of $a$.
   • If $a > 0$, the parabola opens upward, and if $a < 0$, the parabola opens downward.

2. Absolute Maximum and Minimum:

   • A quadratic function has an absolute maximum if the parabola opens downward (as seen in the first graph).
   • A quadratic function has an absolute minimum if the parabola opens upward (as seen in the second graph).

3. Axis of Symmetry:

   • The axis of symmetry is a vertical line that divides the parabola into two mirror images. The equation of this line is $x = -\frac{b}{2a}$, where $b$ and $a$ are coefficients from the quadratic equation.

4. Key Features of the Graphs:

   • The first graph represents the quadratic function $p(w) = -0.1w^2 + 40w + 3300$, which opens downward, so it has an absolute maximum.
   • The second graph represents the quadratic function $r(w) = 3w^2 + 24w + 42$, which opens upward, so it has an absolute minimum.

Analyzing the Questions on the Right:

1. First Function:

   • The function $p(w)$ has an absolute maximum.
   • The graph increases to the point (200, 7500) and then decreases.
   • The graph is symmetric about the line $w = 200$.

2. Second Function:

   • The function $r(w)$ has an absolute minimum.
   • The graph decreases to the point (-4, -12) and then increases.
   • The graph is symmetric about the line $w = -4$.

3. General Properties:

   • The graph on either side of the axis of symmetry behaves symmetrically.
   • In the equation of a quadratic function, the greatest exponent is 2.

Would you like further help solving or understanding these properties in more detail? Here are some related questions to deepen your understanding:

Relative Questions:

1. How do you determine whether a parabola opens upward or downward from the equation?
2. Can you find the vertex of a parabola using a different method?
3. What is the significance of the axis of symmetry in quadratic functions?
4. How do you use the quadratic formula to find the roots of these functions?
5. Can you apply these concepts to real-world problems, such as optimization?

Tip: The vertex of a parabola (either the maximum or minimum point) can always be found using the formula $x = -\frac{b}{2a}$.