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The image you uploaded contains a problem where you're asked to determine suitable function equations based on the given characteristics. Let's analyze each part of the problem:

1. **Part (a):**  
   - The graph has a high point at \( H(1, 7) \) with a low point at \( T(3, 1) \).

2. **Part (b):**  
   - The graph has a local minimum at \( P(5, 1) \), tangent with slope \( -1 \) at \( x = 7 \), and an inflection point at \( W(9, 1) \).

3. **Part (c):**  
   - The inflection segment of the graph has a slope of \( \frac{1}{2} \) with \( Q(6, 3) \). The high and low points are 2 units away from the vertex.

4. **Part (d):**  
   - The graph has a saddle point at \( S(2, 0) \), low point at \( T(2, -2) \), and a high point at \( H(4, 4) \).

### Let's go through each part to determine the function equations:

#### Part (a):
This describes a parabola with a maximum and minimum. A quadratic function can be used:
\[ y = ax^2 + bx + c \]
You can determine the coefficients by substituting the points \( (1, 7) \) and \( (3, 1) \).

#### Part (b):
This suggests a cubic function since it describes a local minimum, tangent behavior, and an inflection point:
\[ y = ax^3 + bx^2 + cx + d \]
Using the conditions \( P(5, 1) \), slope at \( x = 7 \), and inflection at \( W(9, 1) \), you can find the coefficients.

#### Part (c):
This could also be a cubic or higher degree polynomial that satisfies the inflection point with a slope condition and is 2 units away from the vertex.

#### Part (d):
Given the saddle point, this is likely a cubic function:
\[ y = ax^3 + bx^2 + cx + d \]
Saddle point \( S(2, 0) \), along with high and low points, will help in determining the coefficients.

Would you like to proceed with the calculations or any specific part of the problem? Or do you need further details on a particular step? Let me know how you'd like to continue!

Determine suitable function equations based on the given characteristics:

(a) The graph has a high point H(1, 7) and a low point T(3, 1).

(b) The graph has a local minimum at P(5, 1), tangent with slope -1 at x = 7, and an inflection point at W(9, 1).

(c) The inflection segment of the graph has a slope of 1/2 at Q(6, 3), with high and low points 2 units away from the vertex.

(d) The graph has a saddle point at S(2, 0), a low point at T(2, -2), and a high point at H(4, 4).

Learn how to determine quadratic and cubic function equations from key graph features such as high points, low points, inflection points, and saddle points. This problem covers advanced concepts in algebra and calculus, suitable for Grades 11-12.

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