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The question asks us to sketch the graphs of \( f(x - ct) + g(x + ct) \) for three different values of \( t \), based on the provided graphs of \( f(x) \) and \( g(x) \). The given values of \( t \) are:

1. \( t = \frac{a}{2c} \)
2. \( t = \frac{3a}{4c} \)
3. \( t = \frac{a}{c} \)

Let's break this down step by step:

### Step 1: Understanding the given functions
- The graph of \( f(x) \) is a triangular shape centered on the origin, with nonzero values between \( -2a \) and 0. 
- The graph of \( g(x) \) is a V-shaped function, also centered on the origin, but it is nonzero between 0 and \( 2a \).

### Step 2: What happens when we transform \( f(x) \) and \( g(x) \)?
The expressions involve shifting the functions horizontally by a value dependent on \( t \):
- \( f(x - ct) \) shifts the graph of \( f(x) \) by \( ct \) to the right.
- \( g(x + ct) \) shifts the graph of \( g(x) \) by \( ct \) to the left.

Thus, the final function is a sum of the shifted versions of \( f(x) \) and \( g(x) \). We need to sketch these sums for each value of \( t \).

### Step 3: Sketches for the given values of \( t \)
#### 1. For \( t = \frac{a}{2c} \):
- \( f(x - \frac{a}{2}) \): The graph of \( f(x) \) shifts right by \( \frac{a}{2} \).
- \( g(x + \frac{a}{2}) \): The graph of \( g(x) \) shifts left by \( \frac{a}{2} \).

#### 2. For \( t = \frac{3a}{4c} \):
- \( f(x - \frac{3a}{4}) \): The graph of \( f(x) \) shifts right by \( \frac{3a}{4} \).
- \( g(x + \frac{3a}{4}) \): The graph of \( g(x) \) shifts left by \( \frac{3a}{4} \).

#### 3. For \( t = \frac{a}{c} \):
- \( f(x - a) \): The graph of \( f(x) \) shifts right by \( a \).
- \( g(x + a) \): The graph of \( g(x) \) shifts left by \( a \).

### Final Result
For each case, the sum of the two shifted graphs would need to be sketched. In each scenario:
- The sum will combine the contributions from the shifted graphs in the overlapping regions.
- You should find the pointwise sum of the functions over the domain.

Would you like me to generate a detailed image of the sketch for each value of \( t \), or provide further details on how the shifts and sums work?

Two graphs, one of f(x) and the other of g(x), are presented. Using the information from the two graphs, sketch f(x - ct) + g(x + ct) for t = a/2c, 3a/4c, and a/c. Each sketch should be done separately with its own set of axes.

Learn how to sketch the combined functions f(x - ct) + g(x + ct) for specific values of t. This problem explores graph transformations, including shifting functions and combining them, based on given graphs of f(x) and g(x). Detailed explanations and sketches are provided for values t = a/2c, 3a/4c, and a/c. Suitable for students in Grades 10-12.

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