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 $f(1 - x)$.

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.