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Let's go through the problem step by step:

### 15. a) Graph the function \( f(x) = 3(x - 1)^2 - 4 \).

This is a quadratic function in vertex form. The general vertex form of a quadratic is \( f(x) = a(x - h)^2 + k \), where:
- \( (h, k) \) is the vertex.
- \( a \) affects the width and direction of the parabola.

In this case:
- The vertex is \( (1, -4) \), because \( h = 1 \) and \( k = -4 \).
- The coefficient \( a = 3 \) indicates that the parabola opens upwards and is narrower than the standard parabola \( y = x^2 \) because \( |a| > 1 \).

### 15. b) What does \( f(-1) \) represent on the graph? Indicate on the graph how you would find \( f(-1) \).

\( f(-1) \) represents the value of the function when \( x = -1 \). This corresponds to a point on the graph where the x-coordinate is -1, and the y-coordinate is \( f(-1) \).

To find \( f(-1) \), substitute \( x = -1 \) into the function:
\[
f(-1) = 3(-1 - 1)^2 - 4 = 3(-2)^2 - 4 = 3(4) - 4 = 12 - 4 = 8
\]
So, \( f(-1) = 8 \), meaning the point \( (-1, 8) \) is on the graph.

### 15. c) Use the equation to determine the following:
1. \( f(2) - f(1) \)

   First, calculate \( f(2) \) and \( f(1) \):
   \[
   f(2) = 3(2 - 1)^2 - 4 = 3(1)^2 - 4 = 3 - 4 = -1
   \]
   \[
   f(1) = 3(1 - 1)^2 - 4 = 3(0)^2 - 4 = -4
   \]
   Now subtract:
   \[
   f(2) - f(1) = -1 - (-4) = -1 + 4 = 3
   \]

2. \( 2f(3) - 7 \)

   First, calculate \( f(3) \):
   \[
   f(3) = 3(3 - 1)^2 - 4 = 3(2)^2 - 4 = 3(4) - 4 = 12 - 4 = 8
   \]
   Now substitute into the expression:
   \[
   2f(3) - 7 = 2(8) - 7 = 16 - 7 = 9
   \]

3. \( f(1 - x) \)

   Substitute \( 1 - x \) for \( x \) in the original function:
   \[
   f(1 - x) = 3((1 - x) - 1)^2 - 4 = 3(-x)^2 - 4 = 3x^2 - 4
   \]

This is the function \( f(x) \) expressed in terms of \( x \) as \( f(1 - x) \).

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Would you like further details on any part? Here are some additional questions related to this topic:

1. How does the coefficient \( a \) in \( a(x-h)^2 + k \) affect the graph's shape?
2. What are the steps to find the axis of symmetry for a parabola?
3. How does the vertex form of a quadratic function differ from the standard form?
4. How would you find the x-intercepts of the function \( f(x) = 3(x-1)^2 - 4 \)?
5. Can you describe the transformations applied to \( y = x^2 \) to obtain \( f(x) = 3(x-1)^2 - 4 \)?

**Tip:** The vertex form of a quadratic function is particularly useful when you need to quickly identify the vertex and transformations of the graph.

Graph the function f(x) = 3(x - 1)^2 - 4. What does f(-1) represent on the graph? Use the equation to determine: i) f(2) - f(1), ii) 2f(3) - 7, iii) f(1 - x).

Learn how to graph the quadratic function f(x) = 3(x - 1)^2 - 4, find the vertex, and evaluate the function at various points. This problem covers key algebraic concepts, including quadratic functions and function evaluation. Suitable for students in Grades 9-11.

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